Road to Trail
When you mainly run roads and sometimes take a little trail without the terrain getting too difficult to get through, you're fine with your road shoes. But once you are getting the feel for it and want to go out exploring a bit rougher terrain or running longer on the dirt trails, it does pay off to invest in a pair of trail running shoes.
Grip, traction, durability
The most significant difference between trail running shoe and road running shoes is the fact that trail shoes will provide more grip and traction on trails. You'll need that because on the trails your running isn't always straight forward. You'll be making side movements, going up and down and therefor you are relying much more on the outsole of your shoes to keep you from slipping. Apart from the above, which is pretty critical, you'll also notice that your road outsoles will be wearing out much faster on the trails, that's why trail outsoles are often made from a bit tougher rubber compound. The same is relevant for the upper materials of trail running shoes; often the materials used are more durable than those of road running shoes.