What do you like best?
The functionality of basecamp is impressive. Intuitive use, easy organization, depending on what type of people you are working with (tech. vs. non -tech), learning curve upon first using it is moderate and thus it is fairly easy to invite new people to projects.
What do you dislike?
If you are inviting people who have no experience with Basecamp, the learning curve could be a bit steeper. Also, you need to set your notifications wisely, otherwise you will be swamped with messages.
Recommendations to others considering the product
I have looked at several other project management tools and this seems to be quite an intuitive to use tool, visually and conceptually easy to understand. If you can get your team committed to using it, it will be an invaluable help in getting thing planned, tracked, refined and completed across different departments or even just within your team. The archive function will let you go back to anything you have done and retrieve it for future projects. A great tool!
What business problems are you solving with the product? What benefits have you realized?
Project Management, communication between different departments, easier workflow, I know where people are at with their projects, or where the project is at. No longer need to skim through emails to find files, etc.