For my "Classic" TV assignment, I watched the Doctor Who story Spearhead from Space. For a more modern show, I watched a few episodes of Third Rock from the Sun. One of the most obvious differences would be the picture clarity. Even with restoration, the UNIT era episodes of Doctor Who don't come close to anything done today.
Spearhead from Space was also serialized. This wasn't that unusual to do back in the day. Another prominent example I can think of is Rocky and Bullwinkle. I think it harkens back to radio, when serialized stories that kept the listener tuning back in with cliffhangers was the norm. Some modern shows, like Lost for example, use cliffhangers and story arcs, but I think that most shows today prefer stand alone episodes.
One of the differences that I didn't expect was just how studio bound third rock was as opposed to all of the location footage in Doctor Who. You would have thought that the older show would have been more studio bound. Part of it might be the genre difference. Sitcoms are less reliant on visuals than action shows are.
The pacing was different as well. The entire Spearhead from Space arc would probably be crammed into a much shorter time frame nowadays. Just look at the new series. Whether this would be to the show's benefit or detriment is debatable. Some classic stories are notoriously saggy in the middle and featured a lot of unnecessary "corridor acting", that is actors running up and down hallways to pad the run time. Still, I sometimes find that the relaxed pace of an old story is a welcome change to a rushed, ADD inducing modern drama.
That turned out to be a lot about Doctor Who and very little about Third Rock from the Sun, didn't it? Well, it's me. What did you expect?