Tabletop RPGs are fundamentally evolutions of games like Cops n' Robbers or House, or perhaps even like playing with dolls or action figures. The difference is that they're actually games. Cops n' Robbers isn't really a game. You're playing it but it's not a game. There's no rules saying what you can't do. You can decide that you're a SWAT officer in that game if you want. You can even be Robocop. I did once, when I was a kid. He's a cop, right? So that's fair!
Of course, your friends think you're cheating when you play Robocop in Cops n' Robbers, but there are no rules saying you can't. Heck, there aren't even rules for how much Robocop or any cop can get shot before he goes down. That's really what separates RPGs from those kinds of games. Rules.
With that, we have really elaborate systems of how our heroes and villains can hurt each other in our games. I personally think that's both OK and lame at the same time. If you compare combat to an opposed check of any other kind, combat is generally of similar complexity. Card games, strategy games, social interactions or racing (foot, horse or otherwise) are similarly complex to crossing swords. In real life, combat ends much faster than these things, mostly because when you stab someone they usually go down right away. In other events you can come back from a deficit and still win.
Combat gets the complicated rules though and other skill checks get single opposed rolls even though in real life, we do way more opposed checks that aren't fighting than we do fighting. I've been in more fights than most people have, and the number of actual fistfights I've been in number less than 20. I've been in more "fights" that were sparring or whatever, but those don't really count. All my real fights were over in under 10 seconds (typically 2-3 punches), except for one that lasted a bit longer.
It seems sort of weird to put that much weight on combat and not put emphasis on non-combat too. If you have this really simulationist combat system (D&D 4e, I'm looking at you) and then have a really barebones skill system, that feels really tacked-on to me.
Let's put it like this. Your character is also a bassist and you jam out a performance. You want to know how good your character is. Some systems, particularly those without detailed skill rules, would simply say that your character just succeeds. Maybe you roll a stat check or something. So anyone with your dexterity (or charisma or whatever the GM decides is the stat for bass playing) could play the bass as good as you. If you're a bassist IRL that really feels hokey. I have much higher dexterity and charisma (he's missing half a finger and he's kind of a jerk) than my dad but I'd be damned if I can play any instrument as well as he can.
So you go a bit further and now you have skill ratings, and you roll your perform check for a set. Except that feels hokey too. Even if you've played Rock Band you'd know that even a single song has easy and hard parts and you might do some parts amazing (even possibly improv something really cool) and other parts you might make a slight error or a big error or whatever. And if you get into a bass battle where you're playing off the other guy and improv-ing stuff everywhere, that has just as much back and forth as something like combat. Probably more so because nobody is immediately dying.
Now being too simulationist is probably bad too, I mean we don't roll for angle of swing for our swords, angles of deflection or stuff like that. In the end it's opposed rolls or rolls against a target number, and really the difference in combat is stuff like movement, defense, or grappling that add to the complexity of fighting.
It's really not fair to the characters that play in non-combat roles (and anyone who has ever played a rogue knows my pain here) to say simply that they do something fine and be done with it. It's important to know what characters can do and what they can't, and how well they can do that thing. Opposed checks might be all that is needed, or we might not need to roll. But the option should be there, and there should be guidelines for what a good roll means.
Let me use a real example here. In our D&D campaign, we're playing E6 and our party is fighter/cleric/bard. I'm the bard. I'm int/charisma so I have a lot of skill points and I'm good at a lot of stuff. I have a huge amount of points invested in knowledge skills -- I don't have any in planes, geography, architecture, dungeoneering, or martial lore, but I can take 10 and get over 15 in any other check (and I can get 14 in the others due to a feat that lets me roll them as trained). The game gives guidelines for what various numbers mean. A 15 means something that's not common knowledge, but not especially rare. A 20 is privileged information, 25 is a secret no one wants to have known, and so on. Sometimes I ask a question and the DM knows the answer in the setting, but he isn't sure whether he wants me to know the information. A skill check says whether or not I know. Sometimes if I really need to know the answer, it means I'll need to do other forms of research like gather information checks or maybe even asking specific NPCs or library research. Knowing the party limits means I can work to find ways around those limits or get lucky and just know a tidbit of info.
The other part of this is that with a rule system for skill checks you really do need to have a guideline for what kind of roll equals what level of knowledge or ability a character has. D20 is pretty good about this. Every skill in the game says what you can do if you can roll X number. Other games don't do this and that is the big mistake. If you don't know what you can get with a roll of 50 (or whatever), the GM will just match the difficulty to the party's average skill. I was once in a D&D 3.5 game where there were DC30 spot checks to see things 15 feet away that were not actively hiding and in good light. What the hell?
Of course, sometimes you don't need to roll. I haven't rolled a disguise check since the first month or so of playing, because my bonus is so high and I've never impersonated anyone specific. Our DM just decided not to bother having me roll. Sometimes I don't have to roll diplomacy checks if I roleplay really well. If something is obviously trivial for a character or the DM really wants the character to know the info, we don't have to roll.
That's really the thing, though. There's always a time where you don't want players to roll because you want them to succeed, and rules don't stop you from letting that happen. They just provide context for all the blurry stuff in the middle.