The .pyc file is Python that has already been compiled to byte-code. Python automatically runs a .pyc file if it finds one with the same name as a .py file you invoke.
"An Introduction to Python" says this about compiled Python files:
A program doesn't run any faster when it is read from a ‘.pyc’ or ‘.pyo’ file than when it is read from a ‘.py’ file; the only thing that's faster about ‘.pyc’ or ‘.pyo’ files is the speed with which they are loaded.
The advantage of running a .pyc file is that Python doesn't have to incur the overhead of compiling it before running it. Since Python would compile to byte-code before running a .py file anyway, there shouldn't be any performance improvement aside from that.
How much improvement can you get from using compiled .pyc files? That depends on what the script does. For a very brief script that simply prints "Hello World," compiling could constitute a large percentage of the total startup-and-run time. But the cost of compiling a script relative to the total run time diminishes for longer-running scripts.
The script you name on the command-line is never saved to a .pyc file. Only modules loaded by that "main" script are saved in that way.