Bananas and plantains are in fact from the same genus. The former is essentially plantains which are divided up into banana plantains, cooking plantains, or bocadillo plantains, which are the smaller ones you sometimes see.
One of the most glaring differences between the two fruits is that plantains are usually firmer than bananas and they are significantly lower in sugar content.
Plantains are used mostly when they are green, under ripe, or overripe. Plantains which are overripe are used primarily because of how sweet they taste at this stage. Plantains which are either green or unripe have a lot of starch in them and are frequently used in soups and stews.