A lot. Trust me. The other day I flew over my handlebars because I was riding with my tire pressure way too high. Whether your wheel falls off or your shock blows up, the tire almost always takes part of the blame, so it's good to stay aware of what's going on with your rubbers. The biggest revolution in bike tires in the past few years, of course, has been the shift to tubeless—where the tire seals to the rim forming an airtight chamber, making a tube unnecessary and pinchflats a thing of the past. So if you're looking to upgrade your tires, the first thing to consider is whether or not you should also upgrade to tubeless, which might also require new wheels—or at least a kit that will transform your rims.
Tire-wise, it all depends on the type of terrain you ride. In the high, dry, desert Southwest, I require a tire that's beefy enough to handle everything from goatheads to gravel, but since the rocks aren't as sharp as they are in, say, Fruita or Grand Junction, I can get away with higher TPI ("threads per inch"—which makes the tire more flexible, but also more vulnerable to slices). But since I tend to ride more cross-country than downhill, I also need a tire with enough knobby traction, but not so much that it slows me down.
If you hate changing tires as much as I do and want one great cross-country set that does it all, Specialized's Captain Control 2Bliss Ready tires ($45 to $50 per tire; specialized.com) are a good pick. Rugged enough not to get sliced and diced by sharp flora, thanks to Specialized's newest, most durable "Control Casing" construction, and with fast-rolling center knobs that help you corner smoothly and efficiently, these tires make riding more fun that it already is.
Another winner in the all-around cross-country category is the Continental Trail King 2.2 (UST version: $65; conti-online.com). Continental added an additional strip of material above the bead (they call it their "APEX sidewall design"), which stiffens the tire's sidewall, which, in turn, increases stability, durability, and gives you more control on tough turns. Add their "Black Chili Compound" grippy rubber tread which, they say, allows for 30 percent increased cohesion and 5 percent improved mileage. The only downside: Now you have no excuse for not riding faster and in more control.