One thing to be aware of with those types of sites is that they typically give low quality transcodes of the original audio stream. I'd say worse than ffmpeg, but realistically the site itself is probably using ffmpeg on the server-side to do the conversion, but with a much lower quality (To keep the site responsive and reduce server load).
Personally, I've had this requirement before. In my case the solution I devised was to download the video file as an mp4 with a firefox extension (VideodownloadHelper) then extracted the audio from that video file with VLC. (Media->Convert/Save). Certainly more roundabout than ffmpeg.