Considering a shoulder press is usually done on a very high incline or at a 90-degree angle (standing or seated), I'd say that there is quite a difference. As the shoulder press emphasizes the delts (and potentially the upper pecs around the collarbone). However, you can set an incline bench very high as well and get upper pec/shoulder work. Or you can do inclines at a 30 to 45-degree angle and target your chest more so.
Soreness is not an indicator of progress. Soreness is usually due to the fact that the movement is new to the body, or some change was made in the training to elicit a new response. It could happen from switching things up all the time, or it could happen because your nutritional intake is too low causing excessive muscle damage and soreness because the body is not recovering. Typically, you will not be sore over time due to the body adapting. But that doesn't mean you aren't progressing. Do not go by soreness. Do not seek soreness. Seek improvements.
As Charles Staley has said before, "If you want to be sore all the time, become a sparring partner for Mike Tyson."