Russell's paradox is that the set of sets $\{x\mid x\notin x\}$ contains itself if and only if it doesn't.
Cantor's theorem states that for any set $S$, its power set $\mathcal{P}(S)$ has greater cardinality. The proof of this (the one I know about) is by contradiction and goes as follows. Assume $S$ and $\mathcal{P}(S)$ have the same cardinality. Then there exists a bijection $f:S\rightarrow\mathcal{P}(S)$. Now consider the set $R=\{s\mid s\notin f(s)\}$. Since $R\subseteq S$, we have $R\in\mathcal{P}(S)$. Since $f$ is surjective, there must exist some $r\in S$ such that $f(r)=R$. We arrive at our desired contradiction upon noticing: $r\in R\iff r\notin R$.
There is some difference between Russell's paradox and the contradiction in the above proof. In the contradiction, $R$ is a subset of some fixed arbitrary set $S$ whereas in Russell's paradox, we seem to be working in the universe of all sets. In the contradiction, there is this function $f$ distinguishing the element $s$ from its image, whereas in Russell's paradox, we talk directly about whether $x$ is a member of itself.
I'm interested in how significant or profound those differences are.
In the latter 19th century when Set Theory as an area of general study was beginning, it was often assumed that we could assume the existence of the set of all and only those things that had any specific property $P$. This is known as the Axiom Schema of Abstraction. (A "schema" because it it an infinite list of axioms, one for each property $P$ that you can state.) Russell showed this was illogical because the assumption that $\{x:x\not \in x\}$ exists is paradoxical.(Note: It does not depend on any definition of what $\in$ means. Russell offered the Barber Paradox to illustrate this: A barber shaves all those and only those who don't shave themselves. Does the barber shave the barber? For barber, read "set". For shaves, read "contains as a member".)
One remedy was to eliminate Abstraction and replace it with the schema of Comprehension (Specification): Informally it says that if $X$ is a set then there exists a set $Y$ whose members are all,and only, those members of $X$ that have some specified property. The crucial difference is that, although we can say that if a set $X$ exists then $Y=\{x\in X:x\not \in x\}$ exists, we cannot prove from Comprehension that $\{x:x \not \in x\}$ exists.
Comprehension also implies, by contradiction, there is no set $V$ of all sets. Otherwise we would have the set $\{x\in V: x\not \in x\},$ which would be $\{x:x\not \in x\}$ and we'd have Russell's Paradox again.
As already stated in other responses to your Q, Cantor's theorem does employ an instance of Comprehension.
BTW. The original names for some (most?) of the axioms of modern set theory were not English and different writers have at times used different English names for them. "Extensionality" (Informally, sets $X$ and $Y$ are equal iff they have the same members) is also called Regularity... And some textbooks combine the Comprehension schema and the Separation schema into a single schema, which they also call Separation.
I recommend the short introductory text on Set Theory by Suppes.