What is the difference between "filtrated" and "filtered"?

The meanings of jargon terms often have essentially nothing to do with the meanings of the English words they're made from. Nowhere is this more the case than in mathematics.

I would use whichever term seems better established—regardless of whether it sounds artificial to native English speakers who aren't mathematicians—so as to give the reader the best possible chance to figure out what I'm talking about.

Updated: I should just answer your question. To my ear, there is a verb filter, and a count noun filter. There is also a non-count noun filtration. (You can count coffee filters, but the filtration of water through coffee grounds isn't something you can count.) So already the mathematical use of filtration as a count noun ("a filtration") differs from the everyday use.

I am pretty sure I never heard the verb filtrate used in everyday English until I started searching for such uses just now. A Google search for filtrated hits mainly dictionary sites. At the moment, the first non-dictionary hit is a link to this question! I can confirm that to my ear, it's hardly a word, and it sounds weird and artificial. Filtered sounds much nicer to me. It is an actual common, everyday word (and, correspondingly, gets hundreds of times as many Google hits).

The main meaning of filter is derived from the field of chemistry: “to pass (a liquid, gas, light, or sound) through a device to remove unwanted”. It this meaning, filtrate and filter are absolutely synonymous.

In specific uses, such as “people filtered out of the room” or “news began to filter in from the hospital”, it sounds rather weird to use filtrate.

Finally, as a side node, a filtrate (noun) is the name of a liquid coming out of a filter (basically, the filtered or filtrated liquid is a filtrate).

Edit: OK, so there was a second question… which I don't know how to address! I'm not a native speaker, but “filtrated” doesn't sound to weird to me :)

To answer your second question... by that point you're creating technical jargon, so it'll just be whatever people decide on it being. "filtrated K-theory" seems more common than "filtered K-theory", according to Google, so that seems to be winning out, currently.