What's the name of the first point in the first row? 0
What's the name of the first point in the second row? \omega
How is that more complicated or less interesting than cardinals?
The notion of the cardinality of sets comes up everywhere in mathematics, and often enough you end up showing that something holds up to a countable number of exceptions. These two kinds of infinities - cardinality of naturals and reals - are so pervasive, you can't get away from them.
But you can do a lot of math without ever having to deal with the ordinal numbers.
For that reason, cardinals are more interesting to me - not just as a concept in and of itself.
EDIT: Just wanted to add that an order isomorphism has two requirements:
(1) it needs to be a bijection (so order-isomorphic objects have the same cardinality); and
(2) it needs to preserve all inequalities (so a strict inequality among items in one object turns into a strict inequality in the same direction among the corresponding items in the other object).