I'll assume you're talking about the Allen-Bradley instruction sets.
Move (MOV) takes the value from Source and puts it into Destination.
Note that I said "value".
In A-B controllers, MOV automatically converts betwen data types, so you can Move an integer into a floating-point value, and the whole number itself is moved over, with its sign. The bit pattern gets automatically converted from 2's complement integer to 32-bit single precision floating point.
Bit Field Distribute (BTD) is more like a shift register and mask. It takes a specific number of bits from one place in a SINT/INT/DINT tag and moves them to another place in a different tag. Often the group of bits starts at Bit 0, but it doesn't have to.
I notice that you didn't mention COP or CPS (specific to ControlLogix) or CPW (specific to MicroLogix). Those generally work on blocks of data in chunks, and will move bit patterns between different data types without converting them or accounting for the Sign bit.
Because COP/CPS/CPW can be used for differing data types or across element boundaries in a user-defined data type, they require a longer discussion and usually some examples.