The Cloud for PS4 has a lower resistance (~40 ohms) but also a lower sensitivity rating (~95db) than the Cloud Alpha (60 ohms and 98dB). Both numbers together tell you how hard something will be to drive, and the second one tells you how loud the headphones can get with one milliwatt of power.
So on paper the Cloud Alpha is a little louder, but in real-world use I find them nigh identical since the lower resistance of the Cloud for PS4 makes them easier on the somewhat-weak PS4 controller jack. I don't do any kind of volume measurements outside of listening by ear, and I don't have to max either pair out on my PS4 controller to get a comfy volume, but I'm not a high volume listener.
If you want to step up to something louder and stick with a HyperX product in the same price range, you could go for the Cloud II and use its USB amp with your PS4. Or you could check out the Cloud Stinger Wireless. Both of those retail for $99, the same price as the Alpha, and have built-in amplification that will also improve your Dualshock battery life. Using headphones with your PS4 controller cuts its battery run time by 30 to 50 percent depending on volume.