Let go through some things shall we?
Both have acceptable overall stats (Arcanine-555 Houndoom-500) so nothing yet. Both are considered offensive sweepers with powerful move sets. Not much to diverse between them so far. On stats along Arcanine takes it in round one because there isn't massive differences. Arcanine is a bit more on the bulk side and packs more HP compared to Hound.
Well, here it starts to change. Hound normally uses some sort of Sub set, but ingame these are fairly impractical so a nasty plot is a bit harder to pull of. A 4 attack set is viable and sucker punch does beat extreme speed in the priority department, but it doesn't make a whole load of difference considering that Houndoom specializes in...Special attacks. While Arcanine can use Extreme speed much more effectively.
Another upside to using Arcanine is the possibility of mixed sets. So you don't have to use just recoil moves that are annoying to use in-game as you spend all your hard worked cash on more Hyper potions. Give it Flamethrower! Houndoom can use Mixed sets but not as well at the latter.
OK a few things with this one. Arcanine evolves using a stone. This means that you will have to keep it as Growlithe until you have all the right moves for it to learn or find the TM's. As well as finding an actual fire stone. Houndoom evolves from Houndour at Lv 24, a very modest level.
Both can learn just one field move, strength.
Arcanine's move pool is much more diverse allowing it to counter more threats and use interesting combinations. Houndoom's move pool is also good, but less so than Arcanines.
Based on your current team typing doesn't seem to be an issue. Houndomm is weak to more but resists more. So that shouldn't be too much of an issue.
In conclusion, Arcanine is the one to have. Houndoom is still very good and perfectably usable but not the one I would choose.