As a manager, Kompany has a couple of unremarkable years behind him in Belgium, whereas Appleton has vastly more experience, given the choice, and setting the names aside, I'd probably go for the latter.
On budgets, we know that much of the parachute payments are going to be eaten up by debt repayments, and they appear to have a number of players on significant wages who might be hard to shift, parts of the squad look a bit aged, they're losing a number of players out of contract, they'll presumably free up some cash from player sales, but those players will need to be replaced as well.
It looks to me like they need to bring in 10 - 12 new players, and at short notice (due to uncertainty surrounding relegation) they might not get the best value for money, in terms of which is the better job, I don't think there's that much in it between the two clubs