In the US, a grocery store sells a large variety of food and household items. For example, one aisle is filled completely with different types and brands of breakfast cereals. The dairy case will have milk, many types of cheeses, coffee cream, whipping cream, sour cream, eggs, etc, etc.
A convenience store is much smaller than a grocery store, and has a limited variety of household items. They might have one brand of coffee, toilet paper, diapers, things like that. Convenience stores tend to stock a larger variety of soft drinks, beer, wine, snack foods and other "impulse" items. Most convenience stores are open very late or even 24 hours, but their prices tend to be a bit higher than those at the grocery store.