I don't think a racing bike is very practical for commuting-- but it depends what you mean by "road bike". I know my sister-in-law's fancy carbon fiber framed costs-what-we'd-pay-for-a-car bike would not survive my commute. If the frame didn't crack, you'd have to retrue the wheels halfway there. (And I'm not going offroad-- our roads just aren't well maintained.) There's not actually anything wrong with that-- I wouldn't drive a Ferrari on my commute, either.
That said, yes. Faster. (Just like the Fararri is faster) When it comes to top speed, your weight is basically irrelevant: wind and rolling resistance vs. how much power your legs put out determine how fast you can go. So, the lower your rolling resistance, and the more aerodynamic your riding position, the faster you'll go. The skinny road bike tyres do have lower rolling resistnance, but getting high-pressure slicks for a mountain bike or a folder will take away most of that advantage. You'll never be able to crouch down into the super-aero position on a mountain or commuter bike, but you probably wouldn't do that on your regular commute anyway: it's really hard on your back. Especially if you have a backpack, which you will, because a fancy speed bike doesn't sport a rack.
Now, the weight makes a difference for acceleration, and obviously, hill-climbing. Folding bikes are typically fairly light, and some have hub gears which you can shift while stopped-- which will get you off the blocks very nicely indeed. Racing bikes spare every ounce, if you're willing to pay. Mountain bikes are tanks, though, and can take crappy road conditions.
That said, I average 20km/h on hilly, awful bumpy roads on my hybrid bike with the front shocks bouncing almost all the while. (I'm told shocks take 10% of your power, but it's better than being bounced in to traffic.) It's my ideal commuter: the shocks have a quick on/off adjust, so I can get extra speed on smooth sections, the tyres are smooth in the middle, with stubs on the edge-- the stubs never touch the ground unless I start to sink in and need them-- and she's got disk breaks, so when the aforementioned potholes warp my rim, I don't have to fumble with a spoke wrench or disconnect the breaks in order to get home. (which has happened to me before.)
You know what else makes a huge difference, though? The cyclist. I've caught up to and passed many times lycra-suited guys on umpteen-thousand-dollar road bikes that they take out every couple weeks to "race" because I do this shit every day.
(Then we get to the hill-climb, and they kick my ass, but I don't like to talk about that part.)
As for folding bikes? They're not made to be good bikes; they're made to be bikes that fold. There are design tradeoffs that you just cannot escape, and I'd only get one if I really, really needed to squash the bike down for some reason.
tl;dr: Don't drive a Ferrari to work. get a hybrid bike with high-pressure, not-too-knobby tyres. Only get a folding bike if you absolutely need to fold it up for some reason.