I don't know much about it - however the company my wife works for is using a graph database as a people network system. However from what I have seen it still relies heavily on good relational management and table design (plus good data). Perhaps unlike RDBMSs it forces a user to to adopt good practice? The link provided by shadow has a comparison of a complex sql query of 100's of lines v the equivalent 5 or 6 lines in Cipher (the equivalent of SQL). What it doesn't show is what the underlying data looks like. For example the complex sql uses union queries which is typically an indicator of poor design. With good design, the sql statement could be just as short.
That said, graph enables you to add properties to relationships - the sql equivalent would be a separate many to many join table. So may well have a faster operating efficiency as it is intended to handle billions of bits of data. It also claims to be more agile - easy to add in new properties for example. Well, it's easy to add a new field to a table, it's what you do with it once it is there that takes the time.
I suspect where it wins is where the style of the development brief is based on 'I know what I want and will recognise it when I see it' - which lets face it, with the current trend for short term investment is likely to be easier to build on as the world changes more and more rapidly.
In summary, looks like a) an enterprise scale system and b) still requires good relationship management but c) does this in a way that is perhaps a bit more intuitive.