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When reading an article about Frege on Stanford Encyclopedia of Philosophy (https://plato.stanford.edu/entries/frege/#AnaStaNum), in section 2.5 I encountered the following sentence:
But though this defines a sequence of entities which are numbers, this procedure doesn't actually define the concept natural number (finite number).
Does this mean there is a difference between these two concepts? If so, what is the difference?
The point raised in the quote is not the same as the question that you are asking.
In the quote: It is a difference whether we define what one is, and then we define what two is, and so on, or whether we define the abstract concept of (natural) number (as pointed out by Mauro Allegranza in the comments). Of course we can say that natural number refers to the totality of the numbers we have defined, and that is a separate definition, and we may or may not be happy with it. (If you want to learn more about problems with this, look for non-standard models of Peano arithmetic).
Your question: There are many concepts of numbers of some kind that differ from the natural numbers. Examples are integers (including negative numbers), rational numbers, real numbers, complex numbers, (transfinite) ordinal numbers, (transfinite) cardinal numbers, surreal numbers. They have all nothing to do with the point raised in that quote, though.
