Videotaped sitcoms were around for a while. Go watch that rascal Archie Bunker, or Taxi. And several 70’s and 80’s sitcoms were made on tape (and it shows). By the 90’s, most quality sitcoms – and even many crappy ones – were back on film. And yes, even sitcoms shot on film use 3-4 cameras. Like Friends, or Cheers, or Seinfeld. That lets them shoot just one or two takes with a live studio audience and still have lots of camera angles to cut with. Though there are some new sitcoms that use the 1-camera, no live audience approach, like Malcom In the Middle. These are shot more like a typical TV drama, aka Movie Light.
But to answer your question very basically, the “film vs. video” response is totally correct. To help explain a key reason the look is different, here are some details.
Film is a series of 24 individual frames projected in a second.
Here in the U.S. (other countries have different systems), we use the NTSC video format. While the shorthand approach is to say it runs at 30 frames per second, that’s not quite accurate. It actually runs at 60 half-frames per second (known as fields). The lines on your screen are drawn in an interlaced fashion, like you would interlace your fingers. Every other line is drawn, then the tube goes back and draws the lines in between. (This is how CRT screens work… LCD, plasmas, etc have their own techniques, but let’s keep this simple.)
Those 60 half-frames give video a smoother look. More “real-life” but far less glamorous.
Another key factor is that video is an electronic process, and film is a chemical process. Like margarine versus butter, or a frozen dinner versus a gormet cook. One is a cheap imitation of the other but serves its purpose when used correctly. (Video is much cheaper. Requires no processing, so you can shoot something and have it air seconds later like a newscast, etc.)
This answer could go on and on without even getting to new technologies like HD and progressive scan video (a technique where video images don’t interlace), but that should cover the basics.