If you run the VPN yourself then you can set the level of encryption and protocol(s) you want to use. The downside to this is you are usually your own worst enemy in terms of security since not everybody can know everything (the same is true for VPN companies but the bigger ones have a few techs involved so more eyes = more chances of catching a chink in the armor).
Either way, you'll most likely have to provide something identifiable to the VPN or VPS company (name? address? phone number? payment account? IP address? e-mail address? browser/OS information? username?). Sure you can attempt to be 100% anonymous but like I said above, you don't know everything about everything so you'll slip up and reveal something that can be used to identify you somehow.
Now if you want to talk about logging then there's really no guarantee a VPN company doesn't log traffic data or that your VPS provider doesn't have a port sniffer somewhere (with or without their knowledge) so this is a gamble either way.
I guess the real benefit for a VPN company is that they (hopefully) have professionals that know what they are doing and a legal team that knows VPNs like the back of their hand. They most likely have multiple servers and thus multiple IPs so you're not tied to one static IP like on a VPS.
The major benefit of running your own VPN is total control of the protocols and encryption and also not having to share the bandwidth with other VPNs (bad neighbors can impact your browsing and attract unwanted attention to your own traffic). VPSs can be cheaper than most big name VPN services if budget is a concern. Another benefit is you can use your VPS for other things so it has multiple uses and no limitations that VPN services might restrict (i.e. the number of connections allowed, you can create new accounts when you want, no bandwidth throttling, etc...). And if you are using your VPN for something like accessing a secure server, with a VPS you have a dedicated IP so you can whitelist the IP to your servers.
