/ *$/ means 'match zero or more space characters anchored to the end of the string', which will always match since it can always match zero characters. It will also match only space characters - other white space (such as tabs) will not match.

/\s+$/ means 'match one or more white space characters anchored to the end of the string'. The \s includes spaces, tabs, and other forms of white space.

See perlre for more information. The Tutorials section also has several entries in Pattern Matching, Regular Expressions, and Parsing, which you may find helpful.