In one case, you want the replacement expression to be a Perl expression,
In the other, you want the value of $1 to be evaluated.

# s/(10)/$1+1/e
$dollar1 = "10";
$replace = $dollar1 + 1;   # One 'e', so Perl expression.
print $replace;            # 11

# s/(10\+4)/$1/e
$dollar1 = "10+4";
$replace = $dollar1;       # One 'e', so Perl expression.
print $replace;            # 10+4

# s/(10\+4)/$1/ee
$dollar1 = "10+4";
$replace = eval $dollar1;  # Two 'e', so Perl expression + eval.
print $replace;            # 14
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Update: Oops, this was supposed to be a reply to Re^2: /e in regexes.