Where to begin? Well, perhaps with the definitions.
What is the definition of the infinite multiplication the ordinals? What is the definition of the infinite product of cardinals?
Recall that $\omega$ is the least infinite ordinal, so if you have a sequence of ordinals which is increasing, and all its elements are finite the its supremum has to be $\omega$.
For the cardinals, your claim is wrong. The infinite product of the finite cardinals is not $\omega$, but rather $2^{\aleph_0}$. You can use the fact that if $\lambda_i\leq\kappa_i$ for all $i\in I$ then $\prod_{i\in I}\lambda_i\leq\prod_{i\in I}\kappa_i$. Now take $\lambda_i=2,\kappa_i=i$ and $\mu_i=\omega$ for $i\in\omega$. Now we have that $\lambda_i\leq\kappa_i<\mu_i$. Show that all the products have the same cardinality.
The most important tip I can give you is this:
Ordinals are not cardinals. Even if the finite ordinals behave the same in their arithmetics when treated as ordinals, cardinals, integers, rationals, real numbers, complex numbers, and so on; when doing infinitary operations all the rules change, and each system can be discerned from the others.