While they almost mean the same thing, I think "some chicken" sounds better(more polite) than "any chicken". For instance when you have guests over at your place and you are serving them food, I would use "some chicken" instead of "any". However, this might also be something cultural so I would not be surprised if the two are used interchangebly in some Anglophone countries.

'Would you like SOME chicken'? > Ok > to ask if she likes an unspecified amount of chicken (pcs of chciken). 'Would you like ANY chicken'? X > You are asking whether she likes any of the species of chicken ,but its ok unless chicken meat has different varieties that you can chose from, as in "Would you like any sauce (to go with it )?"