Integers are countably infinite, real numbers are uncountably infinite. They are provably different, e.g. by Cantor’s diagonal proof. The number of rational numbers is also countable.
"Countably infinite" is a contradiction in terms. If it's countable, it's not infinite. By definition.
That reals are different than integers does not make either of them countable, or one of a "greater" infinity than the other.
Infinity is not a number. Thus, no arithmetic comparators apply to it. Terms like "greater" or "lesser," "more" or "fewer" are meaningless.