Personally, my kids watch more DVDs than live TV. But they watch on-demand programming (via Netflix) way more than either of those options. So my take on the pros and cons:
Schedule: With live TV, you have to revolve your schedule around the show you want to watch. Dinosaur train comes on at 11:30? Ok, now you have to actually be home at 11:30 to see it. That could be a pro or con depending on your perspective. On the one hand, that can used to teach lessons about time-keeping and boundaries that are beyond their (and your) control. On the other hand, that also means your schedule has to revolve around the TV schedule if you want to see a certain show.
Commercials: That's a big con to live TV in most cases. All the ads for toys and food products and other things they're bound to want. DVD to a lesser degree has advertisements for other DVDs, but not other consumer products and sometimes you can skip them. On-demand programming often has fewer commercials or leaves them out entirely.
Discovery: Live TV makes it easy to discover new shows because they come on automatically after other shows end. Which is good if your child is not excited to try new shows otherwise, but bad if your kid is discovering shows that make you cringe because they are annoying or inappropriate in some way. On demand offers discovery options sometimes as well. Netflix suggests other shows you might like because you watched this or that.
Location: DVDs are more flexible about where they can be watched. You can take them in the car with a portable dvd player. You can take them to grandmas and play them on her tv. You can take them on an airplane where you don't even have wifi to play youtube, etc. In this regard DVDs are a win.
Length of play: while DVDs may have many episodes back to back, if you skip "auto play" mode and use the main menu to select an episode it will almost always play just one episode and then stop automatically. On TV, unless you have a fancy tv with a sleep timer, there's no automatically shut off after this episode option. If you need to play continuously without intervention, TV wins. If you want shorter playback, on demand or DVD wins.