Are there any bike pedals that can be used on road and mountain bikes? (so I can use the same shoes)?

You can use mountain pedals without any problem on road bikes, so you could have the same system on both bikes, just don't show them to your roadie friends ;)

Bike pedals are NOT as "use specific" as some would have you to believe. I'd say use whatever you want.

Why would you think you would necessarily have to use 1 set of clips for mountain biking and a different set for the road?

You can always use egg beaters (Crank Brothers) for any kind of riding. They're especially well suited for mountain biking because mud doesn't get stuck in them as easily, and there's no wrong side to clip in. They come in nice light-weight titanium too for road bikers who obsess over every ounce.

I personally use Speedplay frogs--which are made for mountain biking, but I use them on my road bike. The cleat's also well recessed which is nice for walking around at your destination.

As the others have said there are any number of pedals that will work well with both your mountain and road bike. The typical SPD pedal that you typically find on mid priced mountain bikes is probably the most common and one of the cheaper pedals available.

The main reason people will have different shoes is because of the benefits of each. Comparing a high end road shoe with a high end mountain shoe you will find that the road shoe has a stiffer sole. This allows you to exert more pressure and more pull while riding without wearing out your foot. The mountain shoe because of the nature of the shoe cannot be as stiff, as many times you have to get off the pick and move around you need a shoe with a little flex to it so that its comfortable while walking.