The thing that Basecamp does well is that it is a generic project management tool that can be used for any kind of project - thus its wide appeal. The most interesting alternatives are ones that specialize in particular types of projects. For example, if you are a designer there is Solo and if you are a product manager there is my product Aha! - http://www.aha.io. These alternatives specialize by taking the common elements of project management software (like todo lists, calendars and comments) and tailoring them towards a particular job function.
Totally depends what you're trying to accomplish. Basecamp can be great for managing a whole project in all its complexity. But often times that's overkill, and too many features get in the way.
We[0] differentiate by keeping it as simple as possible: have discussions and make decisions together, that's it.
Other tools are great for other types of projects, such as Trello[1] for a sort of collaborative to-do list.
