Once the initial Flybe hype is out of the way, I'm sure their fares will be around the same as ASW's fares, or else BE won't be able to make any money on the route.
So then it comes down to customer choice. ASW is a great little airline, with smiley crew and no funny business in terms of added charges for bags etc etc. And they fly to loads of other places from NQY, not just LGW, so they deserve (and I'm sure will get) local support.