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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.
Concept of Numbers is obviously very different from numbers themselves.
Its like having employees vs what employees. The decision to have employees - may be your business has grown so much you cannot handle it all yourself and dont want to prevent its growth, may be you want to spend less time working, whatever - is very different from the decision of who to employee, what people to hire.
Parents when take kids out on picnic say, dont count them when they take them and dont count them when they put them back in their vehicle to take them home. School bus drivers do that.
Parents dont use concept of number. If a kid is missing they will call him by name when searching for him. For parents there is no generalizing between the kids, none of them is "replaceable".
For a school bus driver at some level the kids are generalized. He dont know each of the 50 kids in bus by name. At some level, for him the kids are "replaceable".
See, military use numbers all along and rarely names. It shows the intention and the intention shows the level of care. 13th division is very much replaceable with 18th division. They both are supposed to be identical in capabilities as far as military is considered.
There is more to it. Concept of Numbers is also about what system you intend to use. Are natural numbers sufficient or do you want to go in decimals, would fractions be a better choice if you have to go in sub numbers. For example, houses are numbered usually in fractions in my country. You have house number 7/10 say. It shows the intention. The Concept of Numbers used show the intention. You immediately know that the street has total 10 houses in it.
In your business you may decide to use concept of negative numbers in your record of transactions. Its when you decide to note in the book how much you owe others. You may use another concept in which you only record payments and receivals, so you never have to note a negative amount.
