A 64 bit computer can run 64 & 32 bit software but not 16 bit software.
A 32 bit computer can run 32 & 16 bit software but not 64 bit.
Since you were previously running a 32 bit computer, it's possible that some of your existing software is 16 bit. 16 bit has been outdated for quite some time so you may not have any. In that case you have nothing to worry about. However, on the chance you do have some 16 bit software lurking on your existing system, it will not work on a 64 bit computer.
There are ways to make a 64 bit computer handle 16 bit software (such as using a Virtual Machine).