What is the difference between "infinite" and "transfinite"?

I have never properly got my head round exactly what the difference is between "transfinite" and "infinite".

Wikipedia says: transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite.

Huh?

For example, the set of all natural numbers $$\mathbb N$$ is "infinite" in cardinality, in fact "countably infinite" -- but its cardinal $$\aleph_0$$ and the ordinal $$\omega$$ which is the "order type" of $$\mathbb N$$ are defined as being "transfinite".

The rest of that wikipedia article on "transfinite number" does not help too much, except to explain that Cantor coined the term "transfinite" as a sweetener, so to speak, so as to make the medicine of his work go down easier.

But apart from historical reasons to do with battling with hidebound modes of thought (and I am familiar with Cantor's difficulties with Kronecker and those of his way of thinking), does there exist a concrete definition that one can go to that says: "this is what transfinite means: ... and this is what infinite means: ... and the difference between the two is ..."?

The reason being, especially in the non-standard analysis case, that "infinite number" is sort of awkward and can make people think about $$\infty$$ or infinite cardinals somehow, which may be giving the wrong impression. But "transfinite number" sends, to me, a somewhat clearer message that there is a particular context in which the term takes place.