Last edit to add a comment on reliability:
You had me at lever. I own multiple versions of both makes and other's you didn't ask about. I'm not getting into which is better as I'm a lever gun aficionado and appreciate them for the class of weapons they represent.
1) While you only asked about Henry and Marlin, Winchester needs to be in the mix and mentioned because you don't know if you'll stumble across one at the right price in the used rack as you're in the store getting ready to buy one of the aforementioned brands. I do own other brands (but no Rossi’s) so I won't mention them.
I pulled three guns out at random of the brands we're talking about. From the top down: Recent vintage Henry in .44 mag. Middle: Late 90's/early 2000's Winchester AE in .357 mag. Bottom: 1st year (1999?) Marlin Cowboy in .45 Colt.
2) All three, H-M-W shoot equally well. I'm not a target shooter and don't need 'target' accuracy. I shoot straight walled pistol cases in these guns and don't weigh them or my bullets individually. When I was actively hunting big game I needed killing accuracy and these guns will give you that and a little more.
3) I'm surprised how much I like the tube loading of the Henry. Side loading is fine but if you're shooting a lot at the range after an hour or two it gets tedious. Since you're buying this as a hunting gun that's not an issue...and for what it's worth, Henry is now offering side loading gates on their rifles, just not in .357 at this time. One noted complaint about the tube has been tube ejection under heavy hand load recoil. I've not experienced that (and I shoot heavy loads).
Quality: Henry is known for the quality of the wood used in their rifles. My experience with them is they've earned that reputation. However, the bluing (if you can call it that) is somewhat industrial. More a matte but less so appearing then Parkerizing. No luster on the iron framed versions I own. All Marlin and Winchesters I'm currently entrusted with have traditional deep bluing. Again, not a deal breaker for me.
Since there's been a lot of talk about the early vintage "Remlins and QC issues I took these pics of where the stock meet's the receiver on all three brands. Granted, not representative of a current Marlin but gives you an idea of the tolerances at each factory. Of note: all of the recent Marlins I've handled in stores made at their current factory looked fine. I will mention that the standard 1894 Marlin fore end seemed very bulky when compared to the Henry or Winchesters. That does matter to me and you don't realize it until you hold them side by side.
Henry:
Marlin
Winchester
Reliability: I've never had a hiccup from the Marlins or Henry in .327 Mag, .357 mag, .44 mag or 45 Colts. I've never found them to be finicky in regards to overall length of cartridge either (this comment goes for the Winchester as well).
The one issue I've had with reliability in a lever action was with 2 Winchesters and it was the same problem. One I was the owner from new and the other I was the second owner of a very low round count...18 to be exact of the AE shown in this posting. A known problem exists, which when you cycle the lever allows the follow-on cartridge to by pass the cartridge stop at the front of the lever and get lodged in the mechanism under the elevator. That almost always requires disassembly and removal of the lever from the gun to clear. Easily fixed by building up the stop via brazing or welding. There may be more problems but these are the only ones I've ever experienced.
Cycling the levers: both the Marlin and Henry 'fixed' levers (mounted to the frame at the front and rotate down) and can feel smoother to the unfamiliar users than the Winchesters toggle action. No preference for me in that department.
Weight: they all run pretty close depending on what platform (Carbine) and options you're likely to want to add to them. The outlier and what's getting a lot of comments here is the Henry Big boy with the brasslite (sp?) receiver and octagon barrel. That really does add some weight to the overall chassis design. I like shooting them but am not fond of toting that model around.
Take everything we say here with a grain of salt and buy what you want. Ultimately it's your money being laid down and your finger pulling the trigger. Let us know what you buy.