I am not a specialist in the philosophy of medicine, but I know it is a controversial issue. Diseases can be classified by symptoms, or by causes. There are arguments in the philosophy of language (by Kripke and Putnam) to the effect that natural kinds terms do not refer to collections of symptoms. For example the manifestations covered by the term "acid" have change during the history of science, but arguably the term was refering to the hypothetical cause of the manifestations, not to a set of symptoms (taste and so on), and its meaning did not really change. We only got to know better what an acid is. Today there is a precise definition of acid in atomic structural term. The problem is that contrarily to "acid", our terms do not necessarily "cut nature at its joints", they do not always refer to natural kind. So a relevant question would be: do mental diseases refer to natural kinds?
For most of them, they are classified by their symptoms by contemporary psychiatrists (we don't really know the causes, even though in some cases genetic factors are identified). This behavioral approach is a sign that psychiatry is not yet a full fledged science (just as alchemy was classifying natural elements by their aspects, while chemistry has structural definitions). So the question remains open. The question is particularly difficult for bipolarity, schyzophrenia and depression: it's not clear whether these are really different diseases or different aspects of the same type of problem, or not even unified diseases.
Another related question is whether they are actually diseases, or social constructs. Psychological and social phenomena are interrelated in a complex way (think of sects or religious fanatism for example: they are kinds of delusions). This is probably not the case for autism (where genetic factors and brain structural causes have been identified) but again things are more complex for bipolarity schyzophrenia and depression. In any case, if they are diseases, their nature is different from biological diseases: the sense in which an organism is healthy is much more clear than the sense in which a mental is healthy. Definitions of mental health can vary in time and social contexts, and there is probably a continuity with "character flaws" as you mention, or fanatism, etc. Mental diseases are generally defined in terms of suffering by modern psychiatry, but mental suffering is different from physical pain: it is much more subjective (it is controversial whether there is a consistent notion of "pain" covering mental and physical pain) and, again, it is interrelated with social aspects. There are also different non orthodox schools (again a sign that the field is not mature) for example approaches inspired by existantialism and phenomenology which pretend to be less reductive (proposal include relating psychosis to attitudes toward time: depression lives too much in the past, schyzophrenia in the present, bipolarity in the future. E.g. Maldiney).
Concerning the question of responsability in your question: I am not sure it is relevant. In any case we are not responsible for a disease. Diseases, biological or mental, can be innate or acquired (or prevalence can be innate), so the question of whether it is intrinsic to us or can be change is irrelevant too. Perhaps we are more responsible of our character flaws: it's up to us to change our behaviour. To some extent, people can recover from mental diseases too. That does not mean they can be held responsible for their disease in any case.
Here are some more ressources (SEP entries):
If your question is more on the illness/disease distinction: both words have the same translation in many languages ("maladie" in french). The distinction is probably minor and unrelated to the physical/mental distinction.
I found this link, where it is argued that disease refers to a precise cause, whereas illness refers to an unspecified state: http://www.anglaisfacile.com/forum/archives2/forum-anglais-2383.php
In which case the question amounts to whether mental illness can be further specified. My answer below is that it is controversial, in that the field of psychiatry is not yet a mature science.