What do you like best?
This is probably the most stable application server I've ever run. Package deployment on the fly is a cinch, but where it really excels at is integration; if you're running any other IBM software (MQ, DB2, ...) integration is built into it, offering you capabilities you wouldn't achieve with other application servers
What do you dislike?
This server is not for the faint of heart, setting it up can be scary, when faced with all these settings and options. But don't worry, once you get to know and understand them, you'll wonder how you could do without. Also, and this issue is with all IBM products, licensing!
Recommendations to others considering the product
If your business is dependent on running a Java stack to operate, don't cheap out. Yes, the free stuff is great, but think how does downtime, issues, actually cost you. It will far outweigh the cost of running websphere application server.
What business problems are you solving with the product? What benefits have you realized?
We use this as part of our integration layer, running all webservices. We have over 20k transactions going through it on a daily basis, on a single instance of the server.