This might be idiosyncratic, but I tend to use 'hypothesis' for conjectures whose truth is an empirical question, and 'conjecture' for those where that isn't the case. So I'd 'hypothesize' that rigorous testing would show the FreeBSD kernel to have good SMP scalability properties up to 32 cores but perhaps not above that, while I'd 'conjecture' that a particular problem can't be solved in better than O(n log n) in the general case. The first hypothesis is one you could confirm or refute empirically, while the second conjecture is one you could prove or disprove analytically.

In some parts of science 'conjecture' can also be used to give the connotation of a weaker, less grounded 'hypothesis'. This seems to be particularly used in astrophysics, where 'conjecture' sometimes means 'very speculative hypothesis'.

I don't find the distinction that important, though; it's pretty rare that flipping these two terms causes a real misunderstanding.