Rather than thinking of limitations with using triangles (I think this is an over simplification of the topic), you need to understand common workflows that take advantage of quad-dominant meshes.
- adding and removing edge loops (to increase or remove detail)
- using loops to better define features of the mesh (such as facial features, around eyes etc).
- operations on faceloops and edge-loops that speed up selection loop selecting, loop sliding for example.
- operations on grids. ripping & grid-fill for example.
- better deformation with shape keys (a lot could be written on this, read up on mesh topology, facial shapes, deformation).
- better subsurface results (edge loops along important features of the form allow your to use creases too).
However in a specific situation you may find it's not a problem to have a 100% triangle mesh so it doesn't help to be single minded about this.
Instead of defining triangle meshes as bad, try to learn to use these tools and you'll better understand why in most cases, triangle heavy meshes are not ideal for character animation.