Actually, bobn is not right in this case. Your "case 2" is very close to the mark -- the right way would be:

my $change_mode = \&change_mode;
...
# Case 2:  Callback routine called with argument (corrected):
# given that $change_mode is a ref to a sub
$main->Button(-text => 'To Upper Case',
              -command => [ $change_mode, 2 ]
              )->pack;
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The "-command" option on Tk widgets (like the "bind" method) can accept an array ref (anonymous in this case), in which the first element of the array is the subroutine reference, and the remaining elements are args to pass to the sub.

The "bind" case is a little different, because the sub will actually get the widget handle that invoked it as the first arg every time a "bound" event triggers the sub, whereas the "-command" option does not pass the widget handle as the first arg.