.44 Mag on the left, .50 AE on the right.
The .50 is significantly more powerful. I remember an ad for the cartridge claiming more energy at 100 yards than the .44 Mag has at the muzzle. Whether it actually does, I've never bothered to check.
The .50 AE launching those 300+ gr slugs produces a lot more torque than the .44 Mag, and an impressive fireball to boot. The recoil isn't painful, at least to me, but it does let you know you just touched something off. It's actually a lot of fun. Watching those big bullets hit tie plates and blocks of wood is just cool.
The Desert Eagle is generally reliable, but it is prone to limp wrist failures if it isn't manhandled. This is more difficult to do with the torque produced by the .50 than by the .44. If you have the form and/or the forearms to accomplish this, and don't mind the extra cost in ammunition, the .50 has it all over the .44. It is simply more fun.
The .44 Mag Desert Eagle is among the softest shooting .44s on the market. It is much easier to control, and because of this, is likely to end up more reliable for many shooters not accustomed to controlling larger amounts of recoil.