The difference is in the subs' prototypes,

$ perl -Mconstant=CHAR_MAX,127 -e'print defined(prototype "CHAR_MAX")?
+"Yes\n":"No\n"'
Yes
$ perl -MPOSIX=CHAR_MAX -e'print defined(prototype "CHAR_MAX")?"Yes\n"
+:"No\n"'
No
$
[download]

The expression 1 + CHAR_MAX works as expected for either constant or POSIX.