Skinning character (attaching mesh to skeleton)

When you add new mesh to a rigged character, it has to be re-skinned again (to attach the new mesh to that skeleton). Re-skinning with Maya somehow changes that model and old animations I had for the same skeleton with old mesh don't work on this new one inside Unity. What am I doing wrong, where's the problem? Or are there better tools than Maya to skin model quickly?

If you did not already, you should check out this how to:

Try checking for a script called PMheatweight, it's got a much better algorithm for auto-weighting (based on the whole 'bone' lenght) than the standard Maya one, which's purely based on the vertex distance to the joints, as if it's a punctual magnet. Don't expect ideal results for all situations though, tbh I personally prefer starting from scratch.

As for copying existing weights from a model to the other, although there's a weight transfer utility in Maya I suggest you try the 'export weights' and 'import weights' smooth bind menu options, an image-based way of doing it. For that to work you'll need to edit the target model UVs so it roughly matches the original model UVs. Might sound tedious but it's actually much easier than the other methods I've tested.