segment registers were the difference
The intel 80386 introduced configurable size segments, which all the OS vendors used to wet segments to 32-bit and used a single address space, with segment selector registers set to this one large segment. Without much change it was possible to increase address space from 4Gb to 16Gb by using {code,stack,data,extra} registers. All the effort would fall to OS and compiler vendors,
AMD didn't just increase the addressing from 32 to 64 bits, they remapped segment registers as general purpose registers to ensure that AMD64 programs would always be faster than "whatever Intel called their version" (was it x86-64 or ia32-64, can't remember).. Fortunately Intel was persuaded to build AMD64 compatible chips