As has been said, the M77 Hawkeye is the better rifle of the two, but there are significant differences in the designs between the M77 Hawkeye and the American Rifle which may drive a buyer towards the American, even if they favor a finer rifle.
Some of the differences I can recall between the two:
The trigger of the American is user adjustable, while that of the M77 Hawkeye is not. The American trigger uses the Savage-style blade safety in the middle of the trigger, which some people enjoy, and some people loathe.
The barrel of the M77 Hawkeye is a traditional machined shoulder design, while the American uses a shoulderless, nut-style barrel. That means the American can be user/owner rebarreled with minimal tools, and no precision machining required.
The American is a push-feed action with a plunger ejector and a sliding, lug mounted extractor. The M77 Hawkeye is a controlled-round push-feed (meaning it picks up the round from the mag as a CRF, but can also take a round dropped into the chamber, not fed from the mag), with a claw extractor and fixed ejector. This means that feeding in the M77 Hawkeye is more reliable, that extraction more reliable and the extractor more durable in the M77 Hawkeye, and that ejection can be controlled in the M77 Hawkeye by how fast the shooter operates the bolt.
The American is a fast 70* throw, three lug bolthead, while the M77 Hawkeye is a traditional 90*, 2 lug design. This means the American has a shorter and faster bolt lift, but also means the M77 Hawkeye has a lighter bolt lift, and better primary extraction.
Multiple models are available for both, but the standard American Rifle comes with a flimsy polymer stock, while the standard M77 Hawkeye comes with a walnut stock. This difference marks a pretty significant contribution to the price difference between the two, as well as a pretty significant contribution to the aesthetic difference between the two.
The M77 Hawkeye includes integral scope ring bases in the receiver, and the bevel locked and recoil lugged Ruger Ring design. This is an incredibly strong mounting platform, although the positions and ring spacing are fixed, such not all optics will fit for all shooters. The American utilizes a more modern screw-on base design, such the owner has multiple options for one or two piece bases, picatinny rails, RDS mounts, etc, including the dexterity to move the optic fore or aft (almost) however they need.
The standard American Rifle includes detachable box magazines, whereas the M77 Hawkeye is a hinged floorplate, fixed box magazine. For a hunter in most states, this isn’t important, but for someone living in states where rifles cannot be transported while loaded, it can be a hassle. It’s hard to lose a fixed magazine (although I have never lost a DBM either), or hard to forget one when you head to the range (I HAVE forgotten my DBM’s on multiple occasions).
The American uses stock mounted V blocks for recoil transfer, while the M77 Hawkeye uses a more traditional integral recoil lug. The American does use traditional 90* action screws, while the M77 Hawkeye uses a 90* in the rear, and a 45* screw in the front, which resultingly requires proportionately higher torque to generate the same downward force into the bedding/action inlet.
Aftermarket parts are more readily available for the American rifle than the M77 Hawkeye, largely because there’s greater interest to do these modifications to the American (maybe a way of saying owners recognize they NEED upgrades?) and the drop-in compatibility for the American is better than the M77 Hawkeye. Many folks buy Americans to effectively serve as “kit guns,” replacing the stock and doing their own home smithing to make the American look and feel more like the M77 Hawkeye - and spending extra money to do so.
I’m sure I’m forgetting a few things, but those are what come to mind to compare and contrast the two.
Personally, I have no interest in the American Rifles, while I own several M77 MkII and M77 Hawkeyes.