int64_t is a Standard C++ type for a signed integer of exactly 64 bits. int64 is not a standard type.

The first C++ standard didn't have fixed-width types. Before int64_t was added to Standard C++, the different compilers all implemented a 64-bit type but they used their own names for it (e.g. long long, __int64, etc.)

A likely series of events is that this project originally would typedef int64 to the 64-bit type for each compiler it was supported on. But once compilers all started to support Standard C++ better, or once the person who wrote the code found out about int64_t, the code was switched over to use the standard name.

Each type of integer has a different range of storage capacity

Type     Capacity

Int16    (-32,768 to +32,767) or (−2^15 to +2^15−1)

Int32    (-2,147,483,648 to +2,147,483,647) or (−2^31 to +2^31−1)

Int64    (-9,223,372,036,854,775,808 to +9,223,372,036,854,775,807) or (−2^63 to +2^63−1)

As stated by James Sutherland in his answer:

int and Int32 are indeed synonymous; int will be a little more familiar looking, Int32 makes the 32-bitness more explicit to those reading your code. I would be inclined to use int where I just need 'an integer', Int32 where the size is important (cryptographic code, structures) so future maintainers will know it's safe to enlarge an int if appropriate, but should take care changing Int32 variables in the same way.

The resulting code will be identical: the difference is purely one of readability or code appearance.