What do you like best?
I like the philosophy of Lime Text, I have used sublime text in the past for about 2 years. The problems are very clear, it's not open source, there is really just one guy in charge of developing Sublime-Text!
So LimeText being open source will handle this problem. The community will be able to contribute and speed up the development of plugins much quicker.
The best part about LimeText(if you actually get it working), it that it's fast! Today it's main contender is the Atom editor, which in itself is probably the best. But it is relatively slow since it's build using web technology. The backend of LimeText uses the Go programming language. It's much faster, which is the main reason people love SublimeText
What do you dislike?
So far the progress of this project has been very slow. I don't think people are so excited about it and although I think it will be stable at some point, don't expect it to be any time soon.
Recommendations to others considering the product
If you are still using Sublime Text 3 and Love the product, I would suggest to go to Github and star LimeText. In the future you will be able to do everything you can with SublimeText, but you will get faster updates and it will be free :D .
As of today, if you're a working developer, I would just consider you try and play around with it, but that's as far as I'll go.
What business problems are you solving with the product? What benefits have you realized?
The main goal of my text editor is to be able to just get on my computer and instantly be able to make edits to my source files. Be it log files or just quick PHP files. LimeText surely has the speed needed for this.