Combining several objects or animations into one will require slightly less computations (avoiding duplicated initializations, updates, etc), and will therefore have a slightly better performance.
Regarding quality, whatever solution you use, you are able to produce the same level of quality (at the expense of slightly worse performance for example if you use several separate objects).
One final aspect I think you should consider is the maintainability of your solution. If splitting your big object in several small ones makes your iteration time much faster, you should choose this option in the first place (and worry about performance only if it really appears necessary later on).
I realized you might mean you are interested in specifically comparing these cases:
- Using a skinned model with bone animation data created in an external tool
- Using separate rigid models and animating then relatively to each other at run-time procedurally
As noted in the comments, depending on the complexity of your model and on the number of instances you will create, the answer will vary. Try to profile both solutions in your specific scene to find which one performs better.
Regarding quality, it also depends on the kind of model you are talking about, and on the visual style of your game.
If you aim for realistic motion-capture based character animation, I think it will be a lot of work to reproduce it procedurally, and you might prefer the "skinned model" solution.
If you are talking about animated environment models (a "merry-go-round" would be a good example), it should be easy to simulate as separate models.