I have both Roku and chromecast. By far roku offers more choices. chromecast can be controlled by your phone, no pc needed, but if you have a pc anything on your chrome browser can be cast. Casting is very easy both from phone and pc and more apps will utilize the cast feature. Roku can provide you with many more channels to view, plus with a good media server, files like pics and movies can be streamed from your pc, that is not available on chromecast yet. Roku 3 comes with a separate remote that allows you to plug in headphones, has an Ethernet port, and micro SD card slot. Bottom line for $30 (amazon) more why not get both?
Both require Wifi and I would expect a pc is in the mix with wifi. However once set up, I think, not real sure both can be used without pc, as in traveling and hooking up to a tv, like in a hotel. For sure that can be done with chromecast and a smartphone.