Most of your concerns are exclusively about video capture. The higher the resolution (how many pixels wide and tall your video has), the higher the frame rate (how many frames per second your video uses), and the lower the compression (how many slightly different colors are grouped in the same bucket as a single color - that's a bit simplistic but conceptually accurate) - then the more data per second you need to save to your memory card.
For still imaging, the only real concern is if you are planning to shoot a large number of images in long, continuous bursts. This is something those who shoot action and sports are concerned with. Shooting still images of performance art is another area where sustainable burst rate is important.
Assuming the card is the limiting factor, the faster your card can write data, the more frames you can take in a burst before the camera begins to bog down as it waits for data to be written to the card so that more room can be made in the camera's internal memory buffer to store the next shot. Once the camera bogs down, the faster your card can write then the faster each succeeding frame can be taken.
For example, I have Canon EOS 5D Mark IV and 7D Mark II cameras. Both of them have a Compact Flash slot and an SD card slot. The CF card slot runs at a faster bus speed than the SD card slot does.
- With the fastest CF cards, the camera can write a little over 100 MB/s (Megabytes per second. Since one byte is made up of eight bits, one MB/s is equal to eight mb/s or megabits-per-second). For the 7D Mark II that means I can shoot around 120 raw images in a 30 second span. For the first three or four seconds the camera fires at the maximum burst rate of 10 frames per second. After that it slows down as each additional frame can only be taken when enough memory in the buffer is made available after an earlier image has been written to the card.
- With the fastest SD cards, the cameras can only write about 75-80 MB/s. This is because the bus speed of the card slot is only UHS-I compliant. There are much faster SD cards available that use UHS-II, but the bottleneck in this case is the speed of the connection between the camera and the memory card in the SD slot. For the 7D Mark II that means I can only shoot about 95 frames in a 30 second span. The camera still fires at the same 10 fps for the initial 3-4 second burst, but then it takes longer between each additional frame as it takes longer to write each image to the memory card.
If I use a slower card in either slot, the burst rate is even more limited. An old CF card I've had for well over a decade can only write at about 15 MB/s. Once it bogs down, it takes well over one second to write each photo to the card! In a 30 second burst, I can only get about 40 images captured. The first thirty frames are still captured in the initial three second burst, but then only ten more additional frames can be taken in the following 27 seconds!