That kind of "moving clouds animation" is a very common one, as it is used routinely to aid in weather forecasting. However, it is not very common to see it in a whole-world geographic projection as the example you show, I'll elaborate later on why that's the case.
Traditionally, a common source of "cloud animations", have been NOAA's geostationary GOES satellites, in operation since 1975, with 17 satellites launched so far and more to come.
If you are looking for visually compelling animations, the latest GOES-16 satellite is capable of producing color images of the Earth's disk every 15 minutes at a resolution of 0.5 km per pixels (at the equator). They look like this:
And you can access images and animations pretty much right after acquisition, here is an example of the 24 hours before I posted this (reduced to a 20% of the size due to Stack Exchange upload size limits):
You can see updated animations at any time on the GOES-East Image viewer.
More traditional, B&W images (infrared band) can be found in the NOAA GOES page, or many other weather data sources. Also, beside GOES satellites there are many other geostationary weather satellites that cover different parts of the globe.
The reason images like the one you posted are rare (whole-Earth images), is because these kind of images are either taken from:
- Geostationary orbits, that can't see polar areas and suffer of a strong degradation of the ground resolution per pixel at higher latitudes, or
- Polar sun-synchronous orbits, like MODIS or Suomi, that allow whole-world composites but using images from multiple passes. Therefore, the full image is composed of images acquired at different times, leading to an ambiguity in the image's time and an impossibility to produce a seamless image, due to the de displacements of the clouds between the different passes.
Worldview is a great source of such multi-temporal composites, and as an example, here is a preview of yesterday's cloud cover by Suomi:
Worldview also allows to produce animations of the cloud cover spanning several months at one day interval. You just have to click on the video icon in the lower left and select the dates interval.
The GOES-East Image viewer stores images of the last four days (with some gaps some times), if you bulk-download them you can create longer animations. As an example, I just did a one of the last four days that you can see here (I do recommend to enable HD resolution).
Another interesting one, that also allow for animated display is the EPIC camera on DSCOVR satellite. Despite that the time resolution is a bit too low for looking at the clouds evolution, its unique perspective from the L1 point (42 times further than geostationary orbits) allow a full view of the illuminated side of the Earth. In addition from the animation in the link above, this video is also worth checking.
You can also find animations created using models instead of real data. That approach allows seamless global coverage at an arbitrarily high temporal resolution. A nice example is the one year long animation (at one hour time steps) of cloud cover and precipitation created from the NCAR's CESM model.