In general, neutral grip (palms facing each other) presses are much easier on your shoulder. This is true of the dumbbell bench press, pull ups, etc. By easier, I mean less stress on the connective tissue (ligaments and tendons), rotator cuff, as well as being more mechanically advantageous. That means you can load it heavier with a neutral grip (palms facing each other) than the open grip (palms facing forward).
As to which is most advantageous, that depends on what you are after.
- Bodybuilders prefer the palms out grip due to the way it builds the shoulder muscles. They also tend to use the 8-12 rep range, which means the intensity is kept below where it can be a problem.
- If you are trying to help build the strength to put more weight over your head, the palms facing each other approach will be the best bet. That allows you to protect the stabilizing musculature and minimize the stresses in the shoulder joint.
Some forms of competition or strength training involve a lot of overhead work. For example, strong man competitions will have both neutral grip presses and open grip presses. For example, log presses are a neutral grip overhead press while axle clean and press is an open grip overhead press.
As a power lifter, I prefer the neutral grip option for just about all my dumbbell pressing needs. As far as musculature emphasized, there are differences. About as much difference as between pull ups and chin ups (palms facing out vs. palms facing you). 90% of the exercise is the same, but that last 10% can be aesthetics or more shoulder stability.
Another option is to rotate the dumbbell as you press. In the bottom, the dumbbell is in the safer neutral position. At the top, the dumbbell is rotated to the open position. This approach allows you to get the best of both techniques: reduced stress on the shoulder and better aesthetics (as well as strengthening the rotator muscles). If you incorporate this approach, start out with lighter weights and higher reps to get used to the movement better.