A client state is one that retains only a shred of autonomy from its protector state. Foreign diplomacy is carried out by the protector state, as is its research (which, is separate but controlled by the protector), and the protector has control over the client's civics. The protector can also make demands on what is produced in each city, and control the client state's units. The cities and units will be on auto, and will revert to auto after each demand of production or unit tasks. A client state is half way between a territory you have conquered and a vassal.
A client state provides benefits over out-right conquest: a client state can be won by diplomacy instead of lengthy war, it can provide you with large territory with less maintenance as the client retains its capitol, the client state is less likely to resist your rule than if it were conquered outright, and none if submitted diplomatically and only has some unhappiness due to desire for independence. A conquered city from a civilization with culture higher than yours has a good chance switching back--a client state does not switch because its cultural autonomy is maintained. Instead it will try to gain independence.