The manual is quite explicit on the difference:
The mem-aref function is similar to mem-ref but will automatically calculate the offset from an index.
To elaborate, the optional argument to those functions is treated differently, in mem-ref the offset is in bytes, but in mem-aref the offset is the index multiplied by the size of referenced type.
Looking at the code this is in fact the only functional difference:
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(defun mem-aref (ptr type &optional (index 0))
"Like MEM-REF except for accessing 1d arrays."
(mem-ref ptr type (* index (foreign-type-size type))))
Although there are compiler macros which implement optimizations if TYPE and/or INDEX are constant.