Do you think this gaming pc will run better than a ps4?

For the price, that is a really good build. Usually I'd prefer 1600 or 1866 RAM, but at that price...go ahead.

One thing I would change is the wifi adapter. The ones that plug into the mobo and have antennaes are better. Look into that.

Other than that, if it were me I'd spend a little more on the mobo for more features and hopefully better quality too.