I was first going to answer the wrong question : "which example of problems are much harder in hypergraphs than in graphs". I was particularly impressed by the difference in dealing with the maximum matching problem in graphs, and the same with hypergraphs (a set of pairwise disjoint edges), which very easily can model coloring, max independent set, max clique...
Then I noticed it was not your question : "what are the root difficulties between the two ?".
Well, to that one I would answer that up to now I haven't seen much common points between graphs and hypergraphs.
Except the name itself.
And the fact that a lot of people are trying to "extend" the results from the first to the other.
I had the occasion to flip the pages of Berge's "Hypergraphs" and Bollobas' "Set systems" : they contain many tasty results, and the ones I found the most interesting had few to say about graphs. For instance Baranyai's theorem (there is a nice proof in Jukna's book).
I don't know much of them but I am thinking about a hypergraph problem right now and all I can say about it is that I don't feel any graph lurking anywhere around. Perhaps we think of them as "difficult" because we are just trying to study them with the wrong tools. I don't expect the graph problems I'm working on to vanish immediately by using number theory (even though it happens sometimes).
Oh, and something else. They are perhaps harder to study because they are combinatorially much.... more ?!
"try them all and see when it works" is sometimes a good idea for graphs, but with hypergraphs one it quickly humbled by the numbers. :-)