The major win is when you want to shard data or have multi master databases. You can shard data in MySQL but it turns into a major pain. If you are doing a lot of writes it is often useful to shard the data across multiple servers, the problem is that if you want to have strong referential consistency while doing this it can be very hard if not impossible look up CAP theorem.
SQL databases have very good consistency but really bad partitioning support, NoSQL databases tend to go the other way. Easy to partition but often what is called eventual consistency. If you are building a messaging site that is ok, for a bank probably not OK.
The plus is that there are now multiple models to how to store data so there is choice in how you implement stuff, while before all you had were SQL databases.
SE Radio has had a few good episodes on this subject.