To use a gaming analogy, one is the DPS and one is the tank. The programmer moves the needle, makes things happen. But someone has to absorb all the attacks, so that they can do their magic.

The job of a programmer is to focus, deeply. The job of a manager is to handle all the meetings, always pick up calls, cut out unnecessary features.

Without a manager, you tend to take on more than you need to, because there's nobody around to reduce the number of things that need to be done, or estimate the timeline properly. Sometimes you need stuff from the client - API access, and so on. This is the manager's job to keep following up.