To answer the question, remember:
When a word is loaded from memory, the electronics puts the bytes into the register in the correct order. Operations (such as addition) inside the processor use the same order. When the register is stored to memory the bytes are written in the same order. As long as the electronics is consistent, either byte order works. Usually you don't need to think about which order is used.
However, when data from one computer is used on another you do need to be concerned. Say that you have a file of integer data that was written by an old mainframe computer. To read it correctly, you need to know (among other things):
Data is sent across the Internet as groups of bit patterns (of course!) Does the byte ordering matter with Internet data?