I'm wondering why I should think about switching from Test::Simple to Test::More.

If you're doing anything non-trivial it will probably help.

What is it that gives Test::More its "Moreness" ?

It gives you more expressive ways to talk about your tests. So instead of saying:

ok ( $foo =~ /bar/, 'contains bar');
ok ( UNIVERSAL::isa($o, 'MyClass'), 'isa MyClass' );
ok( $foo eq 'bar', 'equals bar' );
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you can say

like( $foo, qr/foo/, 'contains bar' );
isa_ok( $o, 'MyClass' );
is( $foo, 'bar', 'equals bar' );
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which is marginally easier to comprehend. You also get more informative test failures, for example if you do:

is( 'apples', 'oranges', 'expecting to find some oranges' );
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you'll get the informative

not ok 1 - expecting to find some oranges
#     Failed test (-e at line 1)
#          got: 'apples'
#     expected: 'oranges'
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rather than just

not ok 1 - expecting to find some oranges
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You'll find the same sort of advantages with other Test:: modules. You get to express you test more precisely and you get better feedback on test failures.