Well to me its really a question of what you want more Accuracy or Power. If Power is the big concern then the 20 ga by far is the clear winner. I've seen a shot placed in the gut of a deer that took it down and the deer didn't get up. But the bad thing about shotgun slugs is the recoil can be a force to battled. So accuracy can suffer. Personally I like the 44 mag. With the right bullet placement its good out to 120-130yds. Its also just seemed to be much more consistant on the shooting. With in a 100 yds though both are good choices. The handi rifles are good. I grew up hunting with a 20 ga hr slugger with a scope and I'd say if you can handle it thats a great gun. I'd say its really a matter of what you want and what you can handle because neither is a bad choice in my opinion for whitetails with in that range. If you want to go a little more than that range I'd go for the 44 mag.
One other thing the 44 mag has going for it that its compatible with the 44 special which is legal in many states for deer if you have a youngster that wants to go with you that may not be able to handle the recoil.