This may come as a shock to the members of this board, but I own a .444 Marlin.
I have a set of AO ghost ring sights on it and a Leupold scout scope.
Contrary to the fac that I named myself after this caliber on every internet forum I use, I seldom shoot the rifle. I bought the rifle on a whim, used, because the price was something I couldn't pass up. I didn't know anything about the caliber and started researching it on the internet. The search led me to the internet gun forum world. Since I was looking for info on the .444 Marlin, I used that as a name.
My rifle is extremly accurate. Amazing accuracy for a lever action rifle. As Fatelvis mentions, the cartridge is a very versitile little number if you handload. I have loads that are almost silent, up to the 300 grain loads. You can shoot heavier bullets, but I have never done so. I have shot both cast and jacketed bullets and both work great. I also own three .44 mag handguns and being able to use the same bullets in four different guns is a plus for me. Factory ammo leaves a lot to be desired in terms of selection and availability.
Among big bore lever action aficionados, the debate between the .444 Marlin and the .45/70 rage eternal. To me, the point is moot. Yes the .45/70 is more gun. It is better capable of shooting very big bullets. But, if you have enough power to easily shoot clear through any animal in North America from any angle with either one, who cares if one has more "power" than the other one ?
If I was looking for a big bore lever action rifle, and I didn't own a .44 mag handgun and never intended to buy one, I would go with the .45/70. But I think the difference is cutting it pretty fine.
I have shot a few .44 Mag lever action rifles. I have nothing against them. It won't come close to doing what a .444 Marlin can do, but if this isn't needed, the .44 Mag is certainly more pratical than the .444 Marlin. Brass is cheaper and more readily available. Factory ammo is cheaper and more readily available. Recoil is significantly less (I have broken two Lyman reciever sights from the recoil of my .444 Marlin). Range is pretty close to the same since both calibers provide a rainbow like trajectory compared to "normal" long range rifle calibers.
If you are a handloader (or even if you are not), this is a great website for info on all three cartridges out of a lever action rifle (or handgun);
http://beartoothbullets.com/ go to the shooter's forum.