Running shoes usually have very flexy soles so your feet can do their job feeling and responding to the ground. For cycling you want something with a very stiff sole. The reason is that your pedals are much smaller than your feet, so if your shoes don't keep your feet flat, they'll be bending all over in ways that feet aren't really meant to be working. Some people can get away with it, others do get foot pain.

You don't have to go all the way to cycling shoes with hard nylon (or carbon) soles if you don't want to, but the stiffer you can find, the better.