In the specific case of x68 to x68_64, the 64 bit program will be about the same size, if not slightly smaller, use a bit more memory, and run faster. Mostly this is because x86_64 doesn't just have 64 bit registers, it also has twice as many. x86 does not have enough registers to make compiled languages as efficient as they could be, so x86 code spends a lot of instructions and memory bandwidth shifting data back and forth between registers and memory. x86_64 has much less of that, and so it takes a little less space and runs faster. Floating point and bit-twiddling vector instructions are also much more efficient in x86_64.
In general, though, 64 bit code is not necessarily any faster, and is usually larger, both for code and memory usage at runtime.