They aren't really directly comparable; they have fundamentally different motivations.
The "infinity" of algebra and calculus is motivated by the idea of considering larger and larger numbers, and deciding what happens "in the limit". It is often insisted that infinity is not a number, and for a variety of good reasons; arithmetic just usually isn't well-defined when you add infinity to the mix. When infinity is defined as a number, in most conceptions (for real numbers, at least), there is either only one or two infinities (sometimes people will merge $\pm\infty$), or $\infty + 1 \neq \infty$, and there are many infinities of the some order of magnitude, as well as $\infty^2$, and so on, numbers of vastly larger magnitude. In the latter case, arithmetic is well-defined.
The cardinal numbers, by contrast, are just a toolkit for describing how large a set is. It does not describe a limit of smaller things; it describes a static collection's behavior with respect to the equivalence relation of one-to-one correspondence. Subtraction is not well-defined, but addition, multiplication and exponentiation are all well-defined. In this scheme, when $\alpha$ is an infinite cardinal, $\alpha + 1 = \alpha$, as we might intuitively expect. However, there is not only one infinite number; $2^\alpha$ is strictly larger than $\alpha$, according to Cantor's Theorem.
One more collection of infinite numbers related to set theory that might interest you are the ordinal numbers. I'm assuming you haven't seen them before, nor are you familiar with well-ordering.
Here's a thought experiment: imagine counting $0,1,2,3\cdots$. You would never reach it, but let's say you speed up the clock so that you finish up all natural numbers in a finite interval of time. Then the next number you'll count is $\omega$ (for the sake of our experiment). Note that you never said $\omega-1$, and in fact, no such number exists. But then you continue, $\omega+1, \omega+2, \cdots$. Speeding up the clock again, you reach $\omega \cdot 2$. In the same way, you get $\omega\cdot3$, $\omega\cdot4$, etc. Those numbers approach $\omega^2$, and then similarly you get $\omega^3, \omega^4, \cdots$, approaching $\omega^\omega$, and it keeps going. Now, all of the numbers I've described have countably many predecessors, and in order to be able to describe an ordinal number exactly in this manner, it would have to be countable. However, it turns out that if we define a collection of numbers in exactly this manner in set theory, where each number is characterized by the set of predecessors, there are, in fact, numbers with uncountably many predecessors, of any cardinality. These are the ordinal numbers. Have a look at the wikipedia articles on ordinals and well-ordering if you're interested.