Mathematics is the art of clever forgetting. The first mathematical breakthrough, numbers, came about when people realized that if you just forgot about whether it was 5 cows + 3 cows, or 5 rocks + 3 rocks or whatever you always got 8. Numbers are what you get when you look at collections of objects and forget what kind of object they are.
When you say "as sets" you mean you're forgetting a lot of information, in particular you don't care about what the names of the elements in that set are or what properties those elements have. As sets the positive numbers and the positive even numbers are "the same" (that is are in bijection) because you can take 1,2,3,... and just rename 1 to 2, and 2 to 4, and n to 2n, and you've just renamed all the elements and got the even numbers!
However, if you want to remember more about these sets, for example that they're not arbitrary sets they're both subsets of the natural numbers, then they become distinguishable. Depending on how you want to measure "size of a subset of the natural numbers" they might be different sizes. For example, a common way to measure "size of a subset of the natural numbers" is by its "density." That is you look at the first N numbers and calculate what fraction of them are in your set, and then take the limit as N goes to infinity (warning for sufficiently complicated sets this limit might not exist). So for your two examples, one has density 1 and the other has density 1/2, which is one way to make precise the intuition that the former is bigger as a subset of the natural numbers (though not as a set) than the latter.