You should use warnings and strict. It will pick up typos such as close FILE when you meant close FH

The whole date generation can be done more simply with sprintf

sub get_cur_time {
    # Use a slice to get just the bits I need
    my ( $Second, $Minute, $Hour, $Day, $Month, $Year, $WeekDay )
      = (localtime)[ 0 .. 5 ];
    return sprintf '%0.2d/%0.2d/%0.4d %0.2d:%0.2d:%0.2d', $Month + 1, 
+$Day,
      $Year + 1900, $Hour, $Minute, $Second;
}
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You also need to test if the line matches $aDate

while( my $line = <FH> ) {
    if ( $line =~ /\Q$aDate\E/ ) {
        print $line;
    }
}
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I put $aDate between \Q and \E so that any character that has a special meaning in the regex is escaped.