You're doing the counts just right (but you may want to translate punctuation before splitting). You only need to manipulate the hash a bit to get what you want. Taking keys of the hash will give you the list of distinct words, which can be sorted by their value in %count:

my @wordlist = sort {$count{$b} <=> $count{$a}} keys %count;
print 'The text contains ', scalar(@wordlist), ' distinct words', "\n"
+;
print "$_\t$count{$_}\n" for @wordlist;
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You can translate punctuation in just one statement by putting all the punctuation characters in the first slot and using the /d switch.