What are the root difficulties in going from graphs to hypergraphs?
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.