What do you like best?
I work at an advertising agency that also manages some of our clients' websites. I am by no means a web developer, but I have to be able to add and remove campaign tracking codes from client sites quickly. I am not familiar with many .php settings and I really hesitate to change any code on a site directly. Tag Manager is like a back door that's a lot easier to get in to. There's a lot less room for error as well, since you're only editing one small piece of code rather the the core site files. So even if you managed to implement the tag incorrectly, you more than likely didn't break the entire site by doing so and it's a lot easier to diagnose and correct your error.
What do you dislike?
It does take some getting used to and experimentation, but that's pretty standard with most of the core Google products. One thing that really does bug me, though, is the way the Trigger "fire on" options are setup. I've been using TM for 3 years and I still have to test whether I need to use "url", "url path" and "page URL". I just choose one of those each time and test it until the tag fires. There isn't a whole lot of documentation (not that I could find anyway) that explains the differences between those. There needs to be some sort of glossary INCLUDED in the program while you're navigating through. Similar to how Analytics has hover-over tool tips that explain what certain metrics mean.
Recommendations to others considering the product
Just keep things organized. It's easy to accidentally create multiple triggers that essentially do the same thing. You really only need one trigger for each particular function and you can use that trigger to fire multiple different tags. I also recommend implementing some naming conventions to keep everything easy to navigate through.
What business problems are you solving with the product? What benefits have you realized?
I do all of our agency's digital advertising and I use Tag Manager to place tracking codes for each and every campaign we run. Analytics does half the job by telling me what actually happened, but Tag Manager is essential for getting that information into Analytics. Our campaigns would be a lot more difficult to track without it.
Also - I don't know where else to put this - the "Features" part of this survey seems to be asking about Analytics, not Tag Manager. You're also using the Analytics icon, not Tag Manager's.....