The difference between the front and side shoulder press whether performed with kettlebells or dumbbells is the emphasis moving from the front (anterior delt) to the side (lateral delt). The elbows are staying in one line and at the front with the front press, with the side press the elbow is moving from the front to the side during the press.
The front press is the movement most people will be more conditioned for as day to day activities would more likely involve this movement than the side press. The side press is usually more prone to injury, this is due to people using the same weight they use for the front press, i.e. thinking that if they can press 24kg at the front they can press it to the side. The side delts are not conditioned for this position, this is not to mean that you should not be doing side presses, as the movement is awesome and definitely one to include in your programming. It does mean that proper progression and conditioning is required.
The best way to progress and condition for the side press is to include the hybrid press in your training, the hybrid press is between front and side. From there you can start moving more lateral (out and to the side) and add more weight. Not properly progressing with the movement or weight will result in shoulder injuries/shoulder pain.
Whatever you do, the side press is not a press variation you should program in high fast reps for unconditioned people.