Nothing of any real importance: whether or not a cumulative goal shows you the target amount due, or just the delta; on each goal page, which tab is open (“commitment”/“stop/pause”/“data”/“statistics”/“settings”), how the datapoints in the data tab are sorted (I’m not entirely sure why you’d want to sort your datapoints lexicographically by their comments, but however you do sort them, Beeminder will remember that the next time you return to that goal’s datapoints.)
Maybe also a few other things I’m missing, but that’s not the point. All this stuff is necessarily ephemeral: if you switch browsers or devices you’re going to start from zero with the localStorage. So it can’t, by its very nature, include anything more substantial than this kind of minor UI detail.
(If you want to see for yourself what it includes, you can go ahead and run in the devtools something along the lines of
console.log(localStorage) and take a look.)