Very significant, depending on exactly which special vs mag loads you compare a huge difference. For example lets compare a fairly typical 44 Special to a very heavy hunting 44 Magnum load:
44 Special 245gr @ 700 fps roughly 6.5 lbs of recoil energy (coming back into your hand), 246 ft/lbs of energy at the muzzle
44 Magnum 300gr @ 1,300 fps roughly 35 lbs of recoil energy, 1,125 ft/lbs of energy at the muzzle.
As you can see the math shows the magnum being about 5x more powerful, with a similar increase in recoil. Not everyone is sensitive to recoil in the same amounts, some aren't bothered by heavy recoiling guns, others are so even though on paper the magnum has a lot more power and recoil it may not bother you.
A more common 44 mag load would be:
44 magnum 240gr @ 1,250 fps, 23 lbs of recoil, 832 ft/lbs of energy at the muzzle
Still 3 to 4 times the energy and recoil of a 44 Special.