Currently, what conditions are known to be sufficient for the SDP relaxation of Max-Cut to be exact?
This question was studied somewhat in the early '90s (before Goemans--Williamson, in fact; note that it was Delorme and Poljak who first gave a poly-time SDP algorithm for Max-Cut, conjecturing that the 5-cycle gave the worst approximation ratio).
Graphs for which the Max-Cut value and the SDP relaxation coincided were called 'exact'. As Dima says, there are not too many classes of exact graphs, with bipartite graphs being one of the main cases. A number of results and examples are given in the paper "The performance of an eigenvalue bound on the max-cut problem in some classes of graphs" by Delorme and Poljak; probably the best place to start looking.
Note that it is unlikely there is an exact characterization, since it was shown (again by Delorme and Poljak, in "Combinatorial properties and the complexity of a max-cut approximation") that deciding if a given weighted graph is exact is NP-complete. To be fair, they state therein that they do not know the complexity of recognizing unweighted exact graphs.
An obvious sufficient condition is that the SDP in question has a rank-$1$ optimal solution. Indeed, the SDP provides you with an upper bound on the MAXCUT value, and then you pay the price of $\alpha=0.878\dots$ (i.e. your optimal value gets multiplied by $\alpha$) rounding it to a rank one (no longer optimal) solution; so if you can skip the rounding, you're done.
25 years ago (when the original Goemans-Williams paper appeared) it was already known that for random graphs $\alpha$ is much closer to $1$, I don't know whether much more is known nowadays.