When your shoulders need more freedom and barbell bench pressing hurts, use the suspended push-up.
Bench pressing with a straight bar locks the shoulders in place, and this may lead to problems. Having restricted movement capacity, you might end up compensating. Since bench pressing doesn't allow much wiggle room for technical errors, this can be a big problem.
The tighter the shoulders and upper body are, the more you'll overcompensate. Performing suspended push-ups (from rings or straps) will create more freedom and natural movement for the shoulders. By not having to be in a "fixed" position, the shoulder blades can move more freely on the ribcage, which can reduce or eliminate pain.
Many consider push-ups more functional than the bench press. If you load this exercise, there's no reason you "have" to continue bench-pressing if it hurts you.
For proper push-ups you need to avoid spinal/lumbar hyperextension. And your abs need to be ON during push-ups, so this exercise can potentially replace your regular plank/anti-extension training too.