First of all, a clarification is necessary: Connectionism and Pattern Recognition are not the same thing. Neural Networks are one pattern recognition method among many, and in particular Random Forests and Support Vector Machines are very popular among machine learning practitioners without having anything to do with the way the human mind works. In fact, for a while in the mid to late 2000s, it was thought that RF and SVM had completely replaced NNets as the dominant machine learning paradigm, and it was only recently that Deep Learning NNets have made a (debatable) comeback.
Even when they were dominant, the most successful model - Multilayer Perceptrons trained by Backpropagation - had vey little in common with real neurons other than the general concept of representing learning problems as networks and nodes. So saying that NNets are mainstream among the AI community is a stretch, and insofar as they are popular, they are only so because they achieve results, not because they are close to the way the real human minds works.
Now for the answers: Connectionism appears in at least two topics of philosophy of mind.
- Connectionism provides an alternative to those who on one hand do not believe that the human mind is just a biological Turing machine (under the general heading of mind-body functionalism/computational theory of mind) but at the same time do not want to subscribe to any form of mind-body dualism. However, this is premised on the idea that neural networks are somehow capable of super-Turing computation, and this itself is a minority view among computer scientists. In particular for neural networks to be able to go beyond the Turing limit, they would have to be implemented on an analogue computer with unbounded precision, otherwise they face the same limitations that any Turing machine faces.
- Connectionism is considered an argument against the idea that the mind has an inherent language built into it form birth - see for example Noam Chomsky's universal grammar concept or Jerry Fodor's Language Of Thought Hypothesis. Theories such as Chomsky's and Fodor's are actually modern versions of the 17th century concept of innate ideas defended by DesCartes and other rationalists, that there is some knowledge (e.g. logic and reason, moral good, etc...) that we are born with beforehand. Opposed to the theory of innate ideas was Locke and other empiricists Tabula Rasa proposal, that the human mind is a blank slate upon which thoughts are gradually imprinted using sense data. Connectionism is an argument against innate idea type theories and in support of blank slate type theories of how the mind functions, and as such falls into the overall rationalism vs empiricism debate.
Keep in mind that the issue in (1) is an ontological/metaphysical question about what the mind is, whereas the issue in (2) is more of cognitive science psychology question of how the mind works, although both are related. In particular, it possible that the mind is ontologically a neural network, but still has some functional aspects that support the Fodor/Chomsky camp against pure empiricists.
Philosopher of mind David Chalmers has a good collection of references for connectionism with regards to questions of philosophy of mind.