The difference between infinity and the smallest number greater than zero is this:

You cannot capture infinity.

Infinite's just a simpler way of bringing into play the concept of "There is a number larger than any specific numbers which you might think of." Because of this, you can never say, and have it carry any meaning or even be provably true/false, something like infinite < 40e100, or infinite + 1 > infinite. Infinite is like oil in water - it always rises to the top, no matter how you try to capture it.

On the other hand, the smallest number greater than zero, if such a beast exists, can be caught very easily:

Now that we've captured it, it's very simple to find it (by using, for instance, a binary search), and, having found it, create a new smallest number greater than zero.

The reason that no such number exists is that as soon as you "find" it, you can create a new smallest number by halving it. Since math is continuous, and not granular (that is, there is always a real number between any inequal two real numbers, no matter how close together they might be), you will never reach a point where you cannot halve.