For the invisible bird I would use either but they do feel different: "I don't see the bird." meaning please point, I haven't found it yet. and "I can't see the bird" meaning It is behind a tree and it isn't physically possible for me to see it.

In an argument I would, like Faldage, use the same distinction. I would also use either for an argument but "don't" would imply a lack of understanding and "can't" an understanding but disagreement as to the validity of the argument.

There is also the difference in vehemence, both for birds and arguments,eg "I don't agree." vs "I can't agree."
cross-thread alert - denotation vs connotation