public
is visible from wherever.
internal
is visible only within an assembly.
You tend to use internal only to protect internal APIs. For example, you could expose several overloads of a method:
public int Add(int x, int y)
public int Add(int x,int y, int z)
Both of which call the internal method:
internal int Add(int[] numbers)
You can then put a lot of sophistication on a method, but "protect" it using facade methods that may help the programmer to call the method correctly. (The implementation method with the array parameter may have an arbitrary limit of values, for example.)
Also worth noting that using Reflection, any and all methods are callable regardless of their visibility. Another "hack" to control/gain access to internally hidden APIs.
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