First, the grammatical difference: 'Grass' is usually uncountable, whereas 'lawn' is countable. So, you might have grass in your garden (or some grass, or a lot of grass or a little grass). With 'lawn', you'd say that you have 'a lawn', or a large lawn, a small lawn, or several lawns if they're in different parts of a large garden. Secondly, the meaning/reality: 'Grass' is a plant. It occurs naturally in all sorts of places. You get grass growing wild in the countryside, for example. 'A lawn' is a man-made garden feature. To have a lawn, you need to shape it, mow it and tend it. All lawns are grass; not all grass is a lawn.

"Grass" is the actual plant while "a lawn" is the whole area where grass is growing. We use them largely interchangeably, though: "to mow the lawn," "to mow the grass," or "to mow the yard." (Brits might say "to mow the garden" but I don't know.) It only really matters if you're talking about a "species of grass" or something specific to the plant, or if you're talking about it in general: "This house has a beautiful lawn," "He works hard to maintain his lawn." Lawns are really important in America, I'm realizing. Also "grass" is kind of old-fashioned slang for marijuana, so that's an important difference.