Mouthparts are a good way of telling bugs from beetles—if you can get close enough—but there is an easier way.
Both groups of insects have two pairs of wings. (Wingless species also occur, especially among the bugs. But let's ignore those for the moment.)
The fore wings of beetles are modified into hard, protective cases (elytra), which you can see in this tenebrionid. At rest, they cover the hind wings. (The elytra are often much reduced, as is the case with this staphylinid. But they still hide the hind wings, which are folded up.) The inner margins of the elytra are usually parallel, forming a neat junction. In flight, the hind wings do all the work.
The fore wings of most bugs are hardened only at the base. In true bugs (suborder Heteroptera, such as this vegetable bug), the wings lie flat against the body. Unlike elytra, one overlaps the other at the tip. (Usually the right covers the left). In leafhoppers and allies (suborder Auchenorrhyncha), the wings are held at an angle, like the sides of a tent. (A similar situation occurs in those aphids, suborder Sternorrhyncha, that possess wings.*)
*And it obviously doesn't in the wingless ones.