What are the root difficulties in going from graphs to hypergraphs? There are many examples in combinatorics and computer science where we can analyze a graph-theoretic problem but for the problem's hypergraph analog, our tools are lacking.  Why do you think problems often become much harder over 3-uniform hypergraphs than over 2-uniform graphs?  What are the root difficulties?

One issue is that we don't, as of yet, have a satisfactory understanding of spectral hypergraph theory.  Please feel free to shed more light on this issue.  But I'm also looking for other reasons which make hypergraphs more difficult objects.
In this question I understand "difficulty" refers not to "difficult to compute," but to "difficult to study."

Graph problems are easier (at least for me) to study since some concepts happen to be equivalent.  In other words, if you want to generalize questions for graphs to those for hypergraphs, you need to pay attention to the "right" generalization so that the desired consequence can be obtained.

For example, consider a tree.  For graphs, a graph is a tree if it is connected and contains no cycle.  This is equivalent to being connected and having n-1 edges (where n is the number of vertices), and also equivalent to containing no cycle and having n-1 edges.  However, for 3-uniform hypergraphs, let's say a 3-uniform hypergraph is a tree if it is connected and contains no cycle.  But, this is not equivalent to being connected and having n-1 hyperedges, nor to containing no cycle and having n-1 hyperedges.

I've heard a main difficulty to prove the regularity lemma for uniform hypergraphs was to come up with the right definitions of regularity and related concepts.

When you want to consider "spectral hypergraph theory," you may try to look into tensors, or into homology if you see a k-uniform hypergraph as a (k-1)-dimensional simplicial complex, from which linear algebra naturally arises.  I don't know which is the "right" generalization for your purpose, or it's possible that neither is right.