A capacitor stores energy in the form of an electric field while an inductor stores energy in the form of a magnetic field. The difference however in a circuit is that the capacitor stores charge and will eventually cause the voltage to drop to 0 and be equivalent, resulting in current slowing down until there is no more potential. A circuit with an inductor on the other hand will do somewhat of the opposite. It will restrict current from flowing by producing a "back emf" which is similar to adding a resistor to the circuit. After a little time, the change in magnetic field after just turning on a switch or connecting the battery will become constant, and the current will increase.
Of course, both of these graphs are exponential, since the more charge you put into a capacitor, the less the voltage, the lower the current. In the RL circuit the less the the change in magnetic field the less the opposition to current flow, which allows the current to increase, giving a slight magnetic field change.