In terms of raw energy, the 357 has the edge, by a fair margin. If we're talking about "the most deadly bullet" for defense against humans, I believe the 357 still comes out on top...800ft/lbs energy in a few high-end loads, where the 44Spl will top out around 550 and most factory loads don't touch that.
HOWEVER, to get peak effectiveness with the 357 you have to go seriously supersonic and run big pressure and recoil.
The 44 (and similar rounds such as 45LC, 45ACP) can get a lot done while staying subsonic and moderate pressure. This gives them a big advantage indoors (you don't blow your eardrums out) and gives a much more controllable recoil feel in any conditions.
When you keep a .357-cal projectile subsonic (most 38+Ps), effectiveness drops to a point far below the big-bore slugs.
When you drive the 357 up past 1,500fps it turns into a raging monster - on both ends. I happen to think there's a tactical advantage there - set one off in a goblin's direction and instead of going "I'm being shot at!", whoever is left starts thinking stuff like "HOLY CRAP WHERE'D THE CANNON COME FROM!?"
Of course, loads like that are just not compatible with the "compact" 357s. You have to shoot those in a gun that's as big as most 44Spls. My 357 is a Ruger New Vaquero, same size/heft as a Colt SAA and a fine platform for either moderate big-bore loads in 45LC or 44Spl, or exploring full-power 357.