It's hard to say. These tales have one common feature - they are very cruel and scary. However, their cruelness differs: in Andersen stories it is a depressing atmosphere, an enormous feeling of sadness, unexpectedly tragic endings. Grimm brothers' tales are bloody, with limbs being cut off and the bad guys killed after sophisticated tortures. It's hard to believe that, in this form, they are already softened - I mean Grimms' works, which are gathered from people who were telling stories (nannies for example), some old papers, traditional folk stories. It was a former way of upbringing : to threaten children.
You must then choose which one do you prefer: imagination of one man - Andersen, or tradition of generations (German people mostly, but not only. There are significant similarities between Grimms' tales and the French stories collected by Perrault long before).
Though I didn't know anything about the existence of censored versions (except from Disney's films, which I prefer), I agree with HBinjection. It's better to know the originals, especially because you're not a child any more. But if you want to read them to some children, you'll have to use your imagination and change them slightly here and there.
, unless they are twelve or around and love bloody adventures.
Personally speaking, Andersen's stories are on higher artistic level, GrimmBrothers' language is much simpler.
As for who was first: Andersen and Grimm Brothers wrote in XIX century. Because Grimms' tales were not made up by them, they were older.
I wouldn't be able to choose between them - I've got both collections!