For several recent years, Microsoft offers a \"NoSQL\" key/value storage, called \"Table Storage\" (http://azure.microsoft.com/en-us/documentation/articles/storage-dotnet-ho
Table Services is mainly a key-value type NOSQL and DocumentDB is (as the name suggests) a Document Type NoSQL store. What you are asking is essentially the difference between these two types of NOSQL approaches. If you shape your research according to this you should be able to get a better understanding for sure.
Just to keep things simple I suggest you consider the differences between how DocumentDB and Table Services are priced. Not only the cost of these services vary a lot from each other but the fact that DocumentDB works on a "provision first" model and Table Services are offered on a pure consumption based pricing might give you some clues on your compare/contrast.
Let me ask you this; why would I use DocumentDB if the features in Table Services well serve my needs? ;) I suggest you to take a look at how the current Azure Diagnostics tooling use Azure Storage Services, how Storage Metrics use Azure Storage on itself to get a sense of how useful Table Services would be and how overkill DocumentDB might be in some situations.
Hope this helps.