Quaternions came into favor to solve the problem of gimbal lock in composed Euler rotation matrices. This happens when you create a rotation which rotates one axis into another, and end up with a matrix that loses one axis (it loses an eigenvector), and you become "trapped" in the new rotated frame and can't ever rotate out of it. Quaternions don't suffer from this problem, but they're also tricky to work with and reason about, and their rotations can become funny when composed - instead of rotating from orientation A to B, they'll go through a C in a very different place. You need to create heuristics to keep rotations looking sane with quaternions. In the games that I've worked on, we took other precautions to avoid gimbal lock in rotations and stuck to Euler matrices. For example, in a flight simulator, you always computed the final rotation matrix for the plane location directly from its heading, pitch and roll, and so, you'd never suffer gimbal lock.
Why stick to Euler matrices and not some better Geometric Algebra or Quaternions? Because it's really easy to interpolate, and think about plain old rotations about a vector, and you can teach anyone to avoid gimbal lock. It's easier to avoid that problem than to train a bunch of junior engineers on higher level math.