As was already said, the difference between a continuous duty and an intermittent duty solenoid (or relay, for that matter) is that a continuous duty device has been designed for just that - continual current flow. In addition, most continuous duty solenoids have two activating coils - one is the "pull-in" coil while the other is the "hold-in" coil. This is because it takes more effort to move the solenoid plunger than to keep it in its "moved" position. There is often some sort of internal switch to open the pull-in coil circuit once full plunger motion has occurred.
The simplest way to understand the difference between a continuous duty and an intermittent duty solenoid application might be to consider the mechanism that the solenoid is intended to operate. Once moved, will the mechanism tend to remain in that position, or will it have the tendency to return to its original (or some intermediate) position? If the mechanism will reliably remain at rest once moved, than an intermittent duty solenoid should suffice. Otherwise, either a continuous duty solenoid or some sort of latching system will be necessary.
Continuous duty solenoids will require more complex control circuitry...