MGS 3's cinematic presentation, character, and narrative alone make it a must play; nevermind the gameplay improvements.
You play as Naked Snake, aka the future Big Boss. The plot is set during the beginnings of the Cold War. I won't ruin any more of the plot for you, because Snake Eater is just simply a story one should experience entirely on their own. The very (very) plentiful cinematics of the game are what Sons of Liberty seriously lacked, and are immensely satisfying this time around. The games overall presentation is very similar to (should I daresay even modeled after) a Bond film (to which there is even a direct reference at some point in the game). It has a Bond-esque intro, and even a theme song named after itself. The music and its cues throughout the game and its cut-scenes are superb, compelling you to know end. Simply put, Snake Eater is much like an interactive movie, except it's better than most movies.
Interesting characters (some with very detailed back stories) are plentiful, and their unique personalities will keep you intrigued throughout the lengthy cut-scenes. And though Snake Eater, like its predecessors, likes to ramble, you'll find things much more interesting and cleaned up this time around, meaning things are much easier to understand, even though there are still some questions to be answered come Guns of the Patriots.
And... what else was there... oh yeah! Gameplay!
Though the Snake Eater experience is worthwhile even for just the story/cinematics, the gameplay is also great. Never has tactical-espionage action been this intense or satisfying. The number of ways you can take care of guards has increased substantially from previous outings, and sneaking around them/taking them out has never been so intense (you may just find your heart racing harder than ever when "stalking" an opponent or hiding just out of sight from them). The camouflage system is an ideal addition to the Metal Gear Solid play style, and works incredibly well (along with adding yet another layer to the MGS gameplay).
CQC (close quarters combat) is very fun to use, and though it isn't perfect, it adds a layer to the game that MGS1/2 really needed. Now you can grab guards at arms length (even if they're facing you), and choke them, interrogate them, slit their throat, or throw them in multiple directions. You can also do a running throw, an effective way of putting a guard to the ground before they can retaliate after spotting you or getting in your way.
Also, Subsistence's 3D camera angle option and the ability to switch back and forth between it and the traditional Metal Gear camera is a superb addition to gameplay. Many argue that it "takes away" from the "Metal Gear" experience, but I say it's very welcome. Playing with the old camera, though sometimes more useful than the new 3D one, can be just plain painful at times. (I really missed it in the final boss fight, where it isn't allowed).
And though gameplay is quite improved from the previous MGS games, the controls still have their issues. At random points, Snake seems to want to kneel upwards, putting himself in plain view of a guard, when all I was trying to do was continue crawling somewhere. He also (often at points of importance or desperation) wants to just stop crawling, even though there clearly isn't an obstacle in his way. Using the game's many weapons can also be a lot more difficult than it should be (especially out of 1st person view), although it's still entertaining and an improvement upon MGS1/2.
The game also suffers from being a bit of a pain in the ass sometimes (much like the other MGSs). If you're not a seasoned MGS vet, you may find yourself constantly getting alerts, resulting in Caution periods that are more drawn out than ever, especially if you left bodies out in the open, or knocked out a few guards trying to hide from them. It can get repetitive, but overall doesn't hurt the experience (although may cause you to swear at the TV screen occasionally). Other things, too, can be piss annoying, like guards randomly seeing through your scientist disguise, randomly knowing that a box in a room is where you are hiding (even though they didn't see you enter the room nor hide in the box, they just start shooting at it), or randomly having a good idea of where you are hidden (in some boss fights this can become evident).
Overall though, Subsistence's package is well worth these little nitpickings, and the Snake Eater story and its characters should be experienced by all who love a good narrative.