Well, you've spelled each individual word correctly but I'm still not sure what you are asking / describing...
Let me over punctuate your sentence and ask you to better describe what you mean…
what are the pros and cons of an… air vacuum one, ran by pump… and… a water vacuum, ran by gravity.
What is an “air vacuum one”
If you are asking about using a pump to suck water out of the tank for cleaning vs. using a siphon to suck water out of the tank for cleaning…
Then using a pump is usually faster, but most pumps are not designed to have debris pushed through their impeller…
I’ve always used a simple siphon to drain and clean tanks… FYI - The greater the distance below the tank’s water surface the open end of the siphon hose is, the stronger the siphon will be… Therefore if you have a tank upstairs… and you start a siphon with a long hose… and throw that hose out the window… the siphon will go MUCH faster than if you held that hose in a bucket next to the tank…