What do you like best?
If I'm looking for a quick snapshot of how my brand's Twitter is performing, Twitter Analytics is somewhat useful. The UX design is simple and clean, and it's nice to see what our top performing posts have been over a given month.
What do you dislike?
Twitter Analytics is not at all robust (and worse, not easily exportable). If you want to get into the details of what's going on on your Twitter account, you'll have to get your analytics from a paid service. Twitter Analytics isn't half of what Facebook Insights was even 5 years ago.
Recommendations to others considering the product
It's a free feature of Twitter, so if you don't need much, it's a great tool! But if you're trying to do any sort of deeper dive into what's working on your brand page, you'll have to look elsewhere.
What business problems are you solving with the product? What benefits have you realized?
Honestly, Twitter Analytics is a bit of a bust. I can't say it alone is solving any real business problems for us as we have to have a separate analytics tool. It's something I check in on every now and then to see the "greatest hits" on our page.