Getting a pointer again instead of a value:
One usually uses pointer arithmetic when they want to get a pointer again. To get a pointer while using an array index: you are 1) calculating the pointer offset, then 2) getting the value at that memory location, then 3) you have to use & to get the address again. That's more typing and less clean syntax.
Example 1: Let's say you need a pointer to the 512th byte in a buffer
char buffer[1024]
char *p = buffer + 512;
Is cleaner than:
char buffer[1024];
char *p = &buffer[512];
Example 2: More efficient strcat
char buffer[1024];
strcpy(buffer, "hello ");
strcpy(buffer + 6, "world!");
This is cleaner than:
char buffer[1024];
strcpy(buffer, "hello ");
strcpy(&buffer[6], "world!");
Using pointer arithmetic ++ as an iterator:
Incrementing pointers with ++, and decrementing with -- is useful when iterating over each element in an array of elements. It is cleaner than using a separate variable used to keep track of the offset.
Pointer subtraction:
You can use pointer subtraction with pointer arithmetic. This can be useful in some cases to get the element before the one you are pointing to. It can be done with array subscripts too, but it looks really bad and confusing. Especially to a python programmer where a negative subscript is given to index something from the end of the list.