What are the practical limitations of a road bike compared to a mountain bike? I grew up on a big box store mountain bike. I want to get a road bike for my commute, but I'm curious about the limitations I'll experience.

I know the benefits of a road bike: lighter weight, less rolling resistance, aerodynamic riding position, etc. 

But I want to learn about the limitations: I'm mainly concerned with the ability to hop up and down from curbs and occasionally riding through some mulch and grass. 

Is a road bike's frame less tough than that of a mountain bike? On my cheap mountain bike, I can (though I usually don't) hop over and down from curbs at full speed. Would I have to be more ginger with the road bike or would curb-hopping be recommended against altogether? Would the drop bars make it difficult to hop over curbs at all? And would a road bike's tires be unsuitable for anything off-road? Any other limitations?
Honestly hopping curbs isn't something that I would consider as an important factor to consider in choosing a bike for commuting. Riding a road bike it's almost always faster to just stick to the road, and if you do way to exit the roadway it's rare that you will have to travel more than a few seconds before there's a driveway. Still, YMMV - I don't know your route. 

Roadbike frames are fine, it's the wheels that will go first. Don't get silly 16 spoke wheels to commute on. 

There are a couple of things to be aware of with a roadbike though:


crossing small rises at acute angles - with skinny tyres when you roll up a small bump at an acute angle (ie. across the edge of a footpath that is raised above the surrounding grass) if you hit it at too fine an angle your wheel will skid off and straighten out whilst your momentum continues over the bump - resulting in a stack. 
bottom bracket/crank clearance is less than on a mtb - be careful pedalling through bumps/corners.