According to Paxton and other standard references, it was cheaper to let the French occupy themselves. While Germany did occupy northern France, the rump state of Vichy provided several advantages.

Those who would be troublemakers in a totally occupied France could flee to this "still French" if non-Republican state. The Resistance did become more of an issue after they occupied Vichy.

It let a neutral pro-fascist country hold the French navy and the colonial empire, to counteract the claims of the Free French. In theory, by keeping the navy separate from the German navy, the British would not attack it. They did exactly that at Mers-el-Kebir before the ink was dry on the French/German armistice.

As it was, the Vichy regime only lasted a couple of years before the Germans moved into the south, too.