I've not owned one of the .357 models, but several friends have, and I've owned a couple of the very similar .44 Magnum models over the years.
They make a fine close-range deer-and-smaller-game hunting rifle. I wouldn't tackle a moose with one, but if you use a well-constructed bullet and place your shot carefully, any whitetail should be fair game, as well as varmints, fur animals like fox and coyote, and even small game (if legal; in my state, you can't use centerfires for squirrels, as they don't want folks shooting up into the treetops with big bullets -- rimfires only, here). Don't plan on shooting at really big critters or hunting out past 200+ yards with them, as the cartridges just don't shoot flat enough or hit hard enough for these to be reasonable goals; but inside 100 yards (where the majority of deer are harvested, IMO), they will work very well.
They are also popular as "truck guns" or just-in-case "emergency" rifles in rural areas. Anywhere you might see a .30-30 in a gun rack or stashed in a car trunk, the .357 lever-action will fill the same need.
With correct loads, they can also serve as personal defense weapons; in fact, there is quite a following of folks who think these are nearly as effective as an "evil assault-type rifle", without the politically incorrect image problems. They don't reload quite as quickly, but it well-trained hands, they can be formidable defense tools.
Finally, they are fun informal target-shooting guns. I call .22 rimfires "plinkers", and bigger guns in the same role "plunkers." The .357 is light and handy, can shoot the shorter and less-powerful .38 Special ammo for training new shooters of for lower-cost practice, and is usually cheaper to shoot than similar guns in real rifle calibers like .30-30s. You can also shoot pistol-caliber rifles on most indoor ranges, where rifle-caliber rifles and carbines may be restricted from use.
"Placement is power" -- seen in an article by Stephen A. Camp
(RIP, Mr. Camp; you will be remembered, and missed)