I was curious about this so I did some research. From what I gather, there are a few reasons. Curiosity has a 32Kbps connection to earth, or if using one of the two martian satellites as a repeater, up to 256Mbps. The connection is only usable for about 16 hours per earth day due to the axis Mars rotates on.
For reference, most people in 1997 had 56Kbps modems to connect to the internet. Video is totally doable on that connection, it’s just an issue of how long it would take to transmit a video. As an example, a 45 second, 720p video at 30 frames a second taken from my camera takes 145MB. Using Curiosity’s top data rate, (256Mbps) and assuming nothing else is using the connection, it would take about 52 earth minutes to transfer.
Regarding the lunar missions of the 60’s & 70’s video. RF travels at the speed of light, and the moon is very close, so video could be viewed live without problem. Because of the distance, from Mars there is about a 14 minute delay in any data sent or received. Even if you used low-datarate video suitable for streaming over Curiosity’s connection, it wouldn’t be viewable live.
Surely some HD video could be transferred over a several day span, and/or transferred in chunks to give priority to more important functions of the rover. From my readings, Scientists in charge see little value in video. I’m guessing it’s because of the relatively little movement there is on mars. For seeing high detail and changes over time, pictures do the job.
A big part of Curiosity’s mission is to look at rocks and analyze drilling samples. Video doesn’t help much with that. One article I read says that they have actually taken video of the scoop shaking some sand, but because of the data usage, photos are preferred.