Why would you need this:

  $s =~ s/^[ \t]+//mg;   # remove leading whitespace from each line
  $s =~ s/^\s+//;        # remove leading whitespace
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when the second line will do everything that the firstline is doing?

Remember that $s is a multi-line string. So the regex mg modifier in the first regex above ensures that each line in the multi-line string has leading spaces and tabs removed from it.

The second regex, OTOH, does not have any modifiers, so it does not apply to every line in the multi-line string; instead, it trims leading whitespace (this time, including newlines) from the front of the multiline string -- trimming multiple blank lines from the front of a multi-line string, for example.