If you don't use a skill system of some sort, you're resolving challenges with a game of "DM May I?" It's a declarative style of resolution, where if you declare the correct action or actions, the challenge is resolved. And if you know how the DM thinks and know how to appeal to (or just agree with) what the DM considers "common sense", you'll do fine.
For emphasis: this is not a bad thing.
Again: this is not a bad thing.
Here's essentially how it works: when you want to resolve a challenge of some sort (such as the above "sneak up on a guard"), you start talking about all of the preparations your character will take. If your DM agrees that these are adequate preparations and your actions make sense to the DM, then you'll succeed.
Indeed, this can work just as well as a defined skill system, if not better. However, it relies extremely heavily on the DM and the players having a high level of agreement between them with respect to "what makes sense". But if you've been gaming with people for thirty years, odds are you have just that, and arguments over whether or not it "makes sense" to blacken your skin with soot that quickly and successfully, and whether or not you're even able to find sufficient soot to do so, or whether it "makes sense" that the fighter was "stupid" for not wrapping his feet in his tunic to pad them better. This undermines the cooperation and agreement that declarative resolution relies upon.
A defined skill system takes this off the table (for the most part), and only relies on the DM to set the circumstances. It can help create the agreement a group needs to resolve challenges because, hey, it's hard to argue with a 15 on someone's character sheet and a 20 on their die roll.
The other difference between a defined skill-system and a declarative resolution system is that in the former, characters become more of an "imperfect lens" for a player than under the latter. When your character's ability to sneak is quantified on your character sheet as "+0", then your character is bad at sneaking, never mind that you, the player, used to be in the Special Forces and know everything there is to know about stealth. When you roll, your character will likely do worse than the cashier from the grocery store's character with a +15 in stealth. But that's okay, because the cashier's character is the sneaky rogue and your character is a fighter. They're different characters with different capabilities, and just because I'm a medical student doesn't mean that my dumb-as-dirt fighter should be a better healer than the cleric with 10 ranks in Heal.
Personally, my preference leans towards the defined skill-system, as I prefer characters with marked differences, and really want to encourage players to play different sorts of characters. However, I also like it when my players are thinking on their feet and are trying to be clever, so my preference in skill systems are ones that grant players bonuses for smart ideas. For old-school games where characters really aren't much more than avatars for players, and explorations of character motivation aren't really part of the goal, this "imperfect lens" isn't really helpful. In those games, it's about challenging the players, whose characters are glorified game-pieces.
Different types of games and groups have different needs and preferences. Even if it's not my preference, a purely declarative method of resolving skill challenges is certainly workable, especially with the right group.