Programming :: C Header Files And Object Files ?

When is it good to use separate translation units and object files and link them into the main C program, and when is it good to include the header files in the main C program? I don't understand if most people include header files or if most people just link in object files and use their contents in the main program. It's sort of a simple question, but it's confusing to me and that's why I need help with it. I sort of don't understand the difference, or if there's really no difference other than the way the final result is achieved, which way is better or preferred, etc...

For example:

Code:

or:

Code:

Simple explanation of the difference? or which one is preferred or better? I've read a little on the ELF format... so is there no difference in the end result? It's just a matter of preference or necessity, and where the information is to begin with?