Use of antispambot with $curauth->email

As the Codex states for the antispambot() function:

Return Values (string) Converted email address.

You have to do one of the following:

echo antispambot( $curauth->publicemail );
// OR...
print antispambot( $curauth->publicemail );

So your full example would look like the following:

if ( ! empty( $curauth->publicemail ) )
{
    echo 'Email <a href="https://wordpress.stackexchange.com/questions/43732/mailto:".antispambot( $curauth->publicemail ).'?subject=Webmail">
    <img src="'.get_bloginfo('template_url').'/images/email_16.png"></a>'; 
}

About the difference between echo, print and (just) return

1. echo and print are just synonyms/alias for each other and actually display something on the screen. The only difference in usage is, that you can’t use echo in combination with return.
2. return just gives something back as a functions output. In combination with print, a function can actually display something in a template, etc.