Streaming from your PC is a much more computationally demanding task, and so it certainly could be responsible for the poor resolution/quality that you're getting.
Like others mentioned above, if you use the iPlayer app on your phone, you're telling the chromecast to use its own built-in app to stream the data from the BBC server, decompress the data and play the video. No different to watching it on any other single device.
If you go via your computer, your computer is streaming the data, decompressing and playing the video, and then re-compressing the video and sending it via your router to be de-compressed again by the chromecast.
Compressing video throws out information, stuff like colour, contrast and detail, so the extra step of compression when streaming via the PC will make your video look worse. Also, this setup is much more sensitive to slow or disrupted internet speeds on your home network, as the chromecast can't buffer any video ahead of time. Dropping the resolution of the streamed video is the usual way to compensate for that, which might be why your resolution is lower than expected.