Do you want the difference between lut and mux? or lut and ROM?
In terms of FPGA, a lut refers to the small programmable devices (really just ROMS) that create the logic in the design.
A ROM would be a memory device made using integrated RAMs or many LUTs.
LUT - stands for Look Up Table, it's a ROM so you have an address that outputs values for each address so you can produce any function of n-inputs (addresses) at the output of the LUT.
A MUX is used to select one input out of 2^n inputs using the n-input select. You can hook the inputs up to constants that produce the same function of an n-input LUT. So a mux can be made to behave just like the LUT.
A Rom/lut is a memory. It has N address locations. The input is an address to output the value stored at the specified location.
a Mux is a logic device. It has N inputs, and a select input that selects which of the N outputs is connected to the output.