If your #1 priority is to fly asap and you can buy the P3P sooner than the P4 due to its lower cost, then I guess go with the P3P. But other than lower cost, the P4 has every advantage over the P3P.
The Phantom 3 Pro is a very capable unit, and has plenty of ability.
Actually besides collision avoidance, the P4 brings very little more.
The videos are both recorded in max 4K video.
The money you save can be put towards an extra flight battery.
I had the same choice and chose a refurbished P3P (and they actually sent me a new drone) and spent the savings on an extra battery and a few other accessories. I have been very happy with my decision.
If you're just learning I would definitely recommend going with a lower end phantom like the phantom advanced. I would stay away from the standard because it has a completely different type of technology than the phantom three advanced or pro. I would stay away from purchasing on eBay and stick with a reputable dealer.
The advantage of moving from a P3S to P3P is most of your accessories will still work, saving you money on new accessories for the new aircraft. If you have extra batteries for the standard, they will work with the pro (P4 batteries are completely different). Props are the same. Camera filters would be a different size if you have them.