What do you like best?
It is so powerful and is packed with default effects, tools, and features. When you finally get the hang of it, the only limit to the creative possibilities is your imagination (and your computer's CPU/GPU).
I've used it for a bunch of stuff from compositing video files, to 3D scenes, to logo animation, and much more.
Now with the implementation of C4D directly into it's core components, you really have to struggle to find something this application can't do
What do you dislike?
I wish it had more node-based editing capability. Currently, the only node-based thing you can really see is a relationship graph between different compositions. That in and of itself is pretty useful at times, but they need to expand the node-based capability further like in adding effects and animation tools. Many of its competitors have adopted this functionality.
Adobe's technical and business support is and has always been some of the worst experiences of my life with regards to troubleshooting and general help. Every time I have called them or posted on their forums, the response is either: non-existent, not helpful, useless, rude, or just plain incorrect. They seriously need to improve it. If the production of their product was not so high-quality and user-friendly, I'm positive they would go out of business because of this.
Recommendations to others considering the product
It's always being updated with industry standard (and leading) tools. Implementing those and keeping them working is a tough job and Adobe does it well. But give them a little slack when it comes to finding and using different aspects of ae.
What business problems are you solving with the product? What benefits have you realized?
Animation, compositing, basically any time we need an advanced and powerful tool, we turn to After Effects.