There is generally less air blown into the outsole of trail shoe. So the shoes tend to be stiffer so that they do not fold around a rock and cause it to stab into your foot. Sometimes the companies will put some other sort of protection other than sturdier rubber between your foot. The Saucony Exodus for example has a layer or Vibram in the midsole. Trail shoes also tend to have a more 'weather-proofed' upper (the part around your foot) which has a tighter weave to keep out dirt and is usually a combination of darker colors... so that mud does not show. They also tend to have a more rugged tread to provide grip in mud and loose dirt.
Do you need trail shoes to run on a trail? No, not really. It really is just about selling another pair of shoes to you. I usually do most of my running on trails in the same shoes I run on the roads with and never really have any problems unless the shoes are really thin and I step oddly on a random rock. If you are just running on light gravel or your typical single-track surface, you're probably fine with the shoes you have.
However, if the trails are nasty like the other poster has mentioned he deals with in PA, then you might want to invest in a pair. This has never been an issue for me, but I've only run on trails in California, the Colorado front range and the Carolinas for the most part. I imagine there might be some volcanic areas or other places where the rocks are sharp, nasty and annoying, making a pair of sturdier shoes quite helpful.
I dunno, I'm very ambivalent about trail shoes. Seems that while some people find them really useful, for the most part they are unnecessary. Hope that helps.