Mostly, no. Let me clarify. There are three speed limits:

1. The speed limit of your camera
2. The write speed limit of your card.
3. The read speed of your card.

When speed #2 is greater or equal to speed #1, the camera operates at its maximum performance. It will shoot at its highest frame rate until the internal buffer gets full and then slows down while offload to memory which lets you get back to full speed the fastest.

When speed #2 is slower than speed #1, the camera still shoots at its full speed until the internal buffer gets full and then slows down. In this case it slows down more and takes longer to offload before getting back to full-speed.

Speed #3 has no impact on camera use but makes things faster when you view and transfer images from the camera. Notably this will depend on the speed of the transferring device like flash card reader.

Video works pretty much the same. The camera is designed to shoot at a certain bit-rate with a certain resolution and frame-rate. It records all frames it is supposed to.

If speed #2 of your card is faster than is needed, then no problem and you get the maximum performance which records video for the longest duration supported by the camera and card file-system.

If speed #2 is too slow, recording still starts exactly the same way except that it is interrupted and stops. Technically a camera could slow down or compress more but that would make for weird videos and I have never seen a digital camera do that. Mobile phones most likely work the same way but I have not tested.

Therefore the benefits of a faster memory card, up to the maximum supported by the camera or reading device are:

1. Faster time to returning to full-speed continuous shooting.
2. Long video recording duration.
3. Faster viewing of images.
4. Faster offloading of images.

You may note that memory card manufacturers have a number of ways to define their speeds: MB/s (Should be specified for both read and write but not always), Speed Class (Has an official definition which guarantees write-speeds), X (1X = 150 KB/s, usually specified for read-speed).

On the other hands, camera makers extremely rarely say anything about the write speeds of the camera. They sometimes mention that they support a certain UDMA or Class speed.