Both espouse world views that are seen as complete by their followers. Both consider opposing views as incorrect. Both tend to impact human psychology in similar ways through creating an 'us and them' mentality. Beyond the arbitrary split as to what field the terms are individually applied to - ideology for politics and religion for faith/spirituality, is there a concrete difference between the two? For example, within social action, a 'just war' a-la Aquinas and a 'just war' a-la Bush Jr., how are the results of action of the theological and ideological principles different?
Excluding superficial differences like those you pointed out (ideologies tend toward political movements and religions tend toward the spiritual), there are indeed differences between the two. Religions are generally classified as subgroups of ideologies, in that they meet all the requirements and perform similar functions to an ideology, but have unique 'properties' unto themselves.
For example, religions often possess a larger set of conditions that a member need (or ought) to meet, particularly with regard to core beliefs, than ideologies do. Now, that is not to say that ideologies such as Nazism, Communism or Marxism do not necessarily have substantial sets of conditions, but the important distinction comes in how these conditions are met. Specifically, in all the ideologies cited, most of their core principles can be derived from a single notion. For Nazism, it was the importance of racial (i.e. white or 'aryan') purity and the need to hinge that on a kind of Social Darwinism that horribly prejudiced those of Jewish faith and descent. However, by and large, religions can rarely ever derive all their beliefs from a single axiom, partly because many beliefs in religious doctrine exist independently (or at least distinctly) of those core axioms. For example, many who worship one among the Abrahamic faiths attest that God fulfils that of "most perfect conceivable being", but it is difficult to see how this relates (non-arbitrarily, and by that I mean not derived purely from the fact that a connection, however tenuous or explained away by religious text, exists) to the Islamic prohibition on eating pork. These beliefs are not inherently inferable from one another. This is one aspect that, at least for the moment, seems to separate a religion from an ideology.
Often, scriptural and foundational texts are more common in religions, too. Again, one could note that many communists (and also marxists) hold Karl Marx and Friedrich Engels' 'The Communist Manifesto' in especially high regard. But these are exceptions, and even then we can observe some differences between these texts and religious ones. The most important difference concerns that of holiness or of universal truth. The Communist Manifesto may be considered one of the most important political texts, at least by people who adhere to its views, but few would consider it the official maxim on political perspective. It is a guide. On the other hand, while there are a few who may consider religious scripture metaphorical or non-literal, most take these tests to be the literal truths of our world. More precisely, removing the human element of 'perception', we can contend that the motivations for each kind of text diverge. Most texts of a more general ideological nature are seeking a method to 'fix' some kind of "deficiency" (which, for what its worth, could be either positive or negative) they believe exists in human society. In contrast, most of religious scripture's goal is to perpetuate the idea that each book is itself the independent truth of our existence. There are other, smaller differences, but these are the primary ones.
In terms of their actions producing different results, this is more of a grey area, more so that it already was. Certain actions, such as mass suicides, have more recently been linked to religions. Except, there is also significant historical precedent for suicides motived by political systems. Many wars have been born from both political and religious struggles. Even many of the (supposed) positive benefits (community, shared belief, etc.) are very similar. Whilst it is obviously not the case that not every resulting acting of an ideology will align with every such action of a religion (or with other ideologies or religions), I'm tempted to say that there is a mostly shared commonality between the results. At least, to the extent that these beliefs involve large collections of people, anyway. This can sometimes make it difficult to understand whether the root 'cause' of the actions is motivated by the ideologies (and religions) or the individual members (I generally adhere to a case-by-case basis, myself).
As to your question about secularism and what it achieves in light of these beliefs, I think it establishes two important points. One is that it states, contrary to some ideologies, that the nature of a person's belief is not conditional on their existing in some set group or state. That is, it applies a single ideological view that bounds over any and all other ideologies (admittedly with the caveat that they adhere to common moral requirements). The other is that no single ideological or religious belief can ever encapsulate the totality of human belief, but that we can attempt to get as close as possible. Think of it like a limit; the more beliefs one accepts as being apart of ideological experiences, the closer we approach that of absolute ideological acceptance, without ever having to completely agree to every ideology (which would no doubt be a total disaster). It's something of a best-case scenario without the negative (and contradictory) effects of the worst-case scenario. However, it can of course be manipulated and abused, especially when deciding which views to 'secularise', so there would need to be more conditions imposed to ensure it doesn't become radicalised.
One final comment about all this: ideologies are not fixed. By this I don't mean that individual people can change their mind with respect to their belief (although they certainly can) but that no description of, say, 'Christianity' or 'Secularism' or 'Communism' will ever really explain the exact beliefs of every adherent to those ideological systems.
What does secularism achieve?
Secularism, as far as I understood it, arose as a response to arguments between different religious traditions and to mediate between them. It was then not thought as being synonymous to atheism, as this was very much a minority view, and not significant politically or socially.
That has now obviously changed. For example, the philosopher Tariq Ramadan spoke about the Jewish, Muslim, Christian and atheist agoras. A properly secular principle needs to now negotiate between them rather than simply identifying itself with the atheist position.
Your collection of motivating comparisons is full of baseless assumptions with immediate and obvious counterexamples.
No ideology considers itself complete, unless it is entirely unhealthy. Communism has everything to say about financial and social arrangements, but does not purport to explain where food comes from or how life arose. The vast majority of religions do consider themselves to be complete approaches to the entire world, that explain science and nature as well as their central subject matter. They capture an entire culture's shape and behavior, at least at its root in some past time, and therefore cover all of life.
Most ideologies do not consider all other alternatives incorrect. Jungianism or feminism swallow up other ideologies whole and use them, unmodified, as examples and tools. One cannot do that if all other ideologies are wrong. Marxist feminism is a thing. Which could not happen if all feminism considered all of Marxism incorrect. Religions, at least until they are subdued by secular culture, do consider themselves entire explanations, which do require alternative explanations to be wrong.
Your third and fourth observations blatantly contradict one another. Religion is used for politics. The fight between Protestantism and Catholicism that controlled European life for a few centuries, was political. And some ideologies are spiritual, radical pacifism and deep ecology clearly have abstract spiritual values, the value of life and the health of the planet, at their core, which would be considered religious, and are huge parts of various religions. Ideologies are not, in fact, primarily political in nature at all. They are about how ideas fit together. Either ideology or religion, or more naturalistic options like language, national character or traditional ownership can all base us-vs-them politics equally. But that is an abuse, not an essential component, of all of them.
War is a political act, and using either religion or ideology to justify it is simply political abuse of whatever ideas come to hand. There is no distinction in this use, but this use is an alien intrusion into either thing. Since a religion claims to provide a worldview it needs to provide a position on war. And a few ideologies mark out territory, like free markets and equal participation, that they consider worthy of defending violently. But the two are not related. Bush Jr. was obviously both a free-market capitalist and a Christian. He probably considered both justifications for his wars equally, and equally little.
Secularism is itself an ideology. It elevates pragmatism in order to make peace between people of various religions within a single legal system. It serves the purpose that it is set to serve. And again, it is not about the opposition. In fact, it takes forms, like the U.S. First Amendment that are not opposed to religion at all, but attempt to negotiate peace with boundaries, and in fact, were put in place by groups of mostly religious people.
This is, indeed, a difficult question not least because neither term is open to essentialist definition. There is no essence of religion or of ideology whatever their mutual relations might be. There are none the less typical features of religion and typical features of ideology.
Conceptually, I'd say, they have generally different allegiances. Both religion and ideology make use of reason and logic; so there is no total antithesis between them but in ordinary discourse they tend to stand in different relations to reason and logic.
"Religion" is an ancient concept. Beliefs in manas, totems, and taboos are probably as old as humankind itself. Religion provides a cosmology or explanation of the meaning of life and the ordering of the universe by answering questions that many feel are beyond the capacities of reason and logic. Religion thus can insulate the believer from the existential fear that life is without meaning, leaving it a virtual absurdity. Consistent with this function, religion also has furnished society with comprehensive moral systems specifying right and wrong and so establishing duties and responsibilities. Traditionally, religion has fulfilled these roles while retaining a group orientation, as community worship illustrates. This community focus has often allowed religion to serve as the glue holding society together, linking senses of past, present, and future with communal institutions and authority.
Note that there is no claim or implication here that religion merely, in a psychologically self-protective way, insulates the believer from the existential fear that life is without meaning, leaving it a virtual absurdity. There is no hint of an implication that religion reduces barely to this.
Reason and logic can and do fulfil roles in religion in connection with natural theology, casuistry, and the hermeneutic interpretation of texts. (A list, not a complete enumeration.)
"Ideology," in contrast, is a relatively new concept, first appearing during the Enlightenment era of the seventeenth and eighteenth centuries. The Enlightenment was a movement of thought and ideas committed to intellectualism and contemptuous of both tradition and authority. The period was notable for its conviction that "right reason could discover useful knowledge" and its faith that scientific principles could be applied to achieve the freedom necessary for human happiness. French philosopher Destutt de Tracy first used the term "ideology" as a short name for his "science of ideas." The concept encompassed the epistemology of "sensationalism," conceived by an earlier philosopher, Etienne Bonnot de Condillac. Sensationalism posited that simple sensation was the sole source of human ideas and mental powers as well as of reflection and instinct or habit. This theory, therefore, was closely aligned with the scientific method and the need to question and test hypotheses before accepting any fact as proven. Given that virtually all religions believe that reality extends beyond the confines of sensory experience, and given that the root concept of ideology emphasized reason, rationality, and sensory experience, ideology from its very inception was biased against religious perspectives and authorities. The liberal philosophy epitomized by the scientific method of inquiry was at odds with the absolutism of the contemporary European religions. (Stanley Ingber, 'Religion or Ideology: A Needed Clarification of the Religion Clauses' [of the US Constitution], Stanford Law Review, Vol. 41, No. 2 (Jan., 1989), pp. 233-333: 278-9.)
This is not to say that religion eschews reason, rationality, and sensory experience but the trio fulfil a different function in ideology from their role in religion.
If we consider the systems of belief and action that are pre-eminently classed as ideological, it's fair to say (or so it seems to me) that Marxism (aka scientific socialism), democratic socialism, liberalism, and market capitalism all see themselves as based crucially on reason, rationality, and rational experience. Even conservatism does so, though in its Burkean and Oakeshottian embodiments it sees them as embodied in institutions and practices rather than in discursive formations.
Instrumental, means/ end rationality is typical of ideology. Religion's use of reason and logic is somewhat different and is indicated above.
The elements of reason and rationality do not fit well into totalitarian fascism but then as I said at the start, ideology doesn't have an essentialist nature and I didn't undertake to enumerate its essential elements. It has typical features, and totalitarian fascism is atypical.