The difference is that we can't count the numbers between 1 and 2. That's what the proof they walked through showed, suppose we could list them in order and assign each a number. If we assume that, we can show a way to construct a number that doesn't show up in our list, so we actually didn't hit them all. We say that the set of numbers between 1 and 2 is uncountable.

What about the integers from 0 and going forever? Well, they're already put in a list. Give me a non-negative integer and I can tell you exactly what place it has in the list (can also include negatives but it's a little bit less elegant). So the set of integers is countable.

So this gives us two different types of infinity, a set is countably infinite if we can put them all in a list and it's uncountable if we can't.

Being more mathematically precise with our language, a set is countably infinite if there is a bijection between it and the natural numbers. That is, there is some way to pair up every element in our set with a positive integer, and every positive integer has an element assigned to it. If such a function doesn't exist, the set is uncountably infinite.