The overhead barbell shoulder press activates the shoulders and triceps through shoulder flexion (raising your arm towards the ceiling), abduction (raising your arm away from the body) and triceps. There is also significant activation of the muscles which stabilise the shoulder blades.
The bench press involves a significant amount of shoulder flexion when pushing the barbell off the chest including high activation of the anterior deltoids. Reya et al. investigated factors that underly bench press performance. Their findings found that elbow and shoulder flexion strength was moderately associated with bench press 1-rep max. As such, doing overhead shoulder pressing movements can improve your bench press strength.
For powerlifters and bodybuilders who suffer from shoulder pain, some lifters may spend time doing low-intensity resistance band movements or external/internal rotation exercises to strengthen the shoulder stabiliser muscles. An issue with low load exercises is they may be too low in intensity to increase the loading capacity and strength of the muscles that stabilise the shoulder. As the overhead barbell shoulder press takes the scapular muscles through a full range of motion and provides high shoulder stabiliser muscle activation, overhead pressing may help reduce shoulder pain. The overhead press can be included in your training program and could potentially be used as a rehab exercise to help rebuild shoulder strength. For those that are suffering from shoulder pain, you should start with a lightweight and increase the weight slowly depending on your pain threshold.
Kroell, Jordan & Mike, Jonathan. (2017). Exploring the Standing Barbell Overhead Press. Strength and Conditioning Journal. 39. 1. 10.1519/SSC.0000000000000324.
Reya M, Škarabot J, Cvetičanin B, Šarabon N. Factors Underlying Bench Press Performance in Elite Competitive Powerlifters. J Strength Cond Res. 2021;35(8):2179-2186. doi:10.1519/JSC.0000000000003097