default_value() is the value that will be put in the variables_map if the user didn’t specify another value:
./a.out # implies width=75 if that's the default_value for width
./a.out --width=80 # default_value not used
implicit_value() is the value that will be used if the user specifies the option but without an adjacent value.
./a.out --width # implies width=75 if that's the implicit_value for width
./a.out --width=80 # implicit value not used
If you use implicit_value then in commandline options’s short options the user must specify the value immediately after the option:
./a.out -w80 # implicit_value not used
./a.out -w 80 # wrong: 80 parsed as extra arg if implicit_value is defined