Not a satisfying answer, but the best I can think of at the moment.
"Solve" for Coth[a h]
In:= Map[a*#&,Map[1/#&,a Tanh[a h]==b Tanh[b h]]]
Out= Coth[a h] == a/b Coth[b h]
Use that result as a replacement rule
In:= b Coth[a h]-a Coth[b h]/.Coth[a h]->a/b Coth[b h]
That is really no better than doing it manually. And it assumes, without checking, that a,b,h are nonzero.