What are the reasons for choosing a seated shoulder press instead of a standing press (barbell or dumbbell)?
Standing press requires more of the stabilizing musculature than with the seated one. It should however be mentioned that a lot of people lack the required mobility in the shoulder joint to perform the standing press without going into lumbar hyperlordosis. While doing so you effectively exclude the musculature on the front of the abdomen and let the back musculature work on overtime stabilizing your body in a poor position. Tightness in the pectoral minor and lack of thoracic extension is often involved in this. Therefore when performing the standing press you should always make sure that your core is tight and that your rib cage does not tilt in order to make up for crappy shoulder positioning. An exercise I like to recommend to loosen up the shoulder is called "shoulder dislocations", which you can read about here: http://stronglifts.com/shoulders-dislocations/
The standing shoulder press is far more beneficial than a sitting one. This is not to say that the sitting one is useless for people with limited mobility or who are trying to isolate just one muscle group.
While standing, your body is able to activate the stabilizer muscles in your back as your arms extend upward.
The use of the back support while doing a seated shoulder press can lead to faulty recruitment patterns in the muscle groups.
If you do plan to use the seated shoulder press I'd advise that you do so without the back support. Remember to use those abs it will save you spine!
Also, the use of the legs in a standing shoulder press is not cheating. It utilizes the entire body and helps develop explosiveness.
It would be better to do the seated bench press instead of the standing press if:
- you have problems with the standing press (based on Wingeer's answer), or
- you want to target the front shoulder without using any other muscles for aesthetic reasons (based on comments by Dave and Affe), or
- you want strong shoulders (based on Mike's answer)