The title says it all. Accumulation point has a widely known definition: a point in $X$ is accumulation point if every open set containing $x$ contains infinitely many points of $X$
Sometimes I browse online and see that $\omega$-accumulation point being used.
What is the difference between these two concepts?
Your "widely known definition" is not actually the definition that I am used to. I would say an accumulation point of a subset $A$ of a topological space $X$ is a point $x\in X$ such that every neighborhood of $x$ contains a point of $A$ that is not equal to $x$. The term "$\omega$-accumulation point" would then refer to your definition: $x$ is an $\omega$-accumulation point of $A$ if every neighborhood of $x$ contains infinitely many points of $A$. The two notions are equivalent if $X$ is $T_1$, so you might find sources which don't distinguish the two (say, because they are only dealing with metric spaces).