No, it's not necessary to convert to timestamp to do the comparison, given that the strings are validated as dates in 'YYYY-MM-DD' canonical format.
This test will work:
( ( $date_from_user >= $start_date ) && ( $date_from_user <= $end_date ) )
given:
$start_date = '2009-06-17';
$end_date = '2009-09-05';
$date_from_user = '2009-08-28';
NOTE: comparing strings allows for "non-valid" dates e.g. '2009-13-32' and for weirdly formatted strings '2009/3/3', such that the string comparison is not equivalent to a date or timestamp comparison.
[ADDENDUM]
If you do go for the php timestamp conversion, be aware of the limitations.
On some platforms, php does not support timestamp values earlier than 1970-01-01 and/or later than 2038-01-19. (That's the nature of the unix timestamp 32-bit integer.) Later versions pf php (5.3?) are supposed to address that.
The timezone can also be an issue, if you aren't careful to use the same timezone when converting from string to timestamp and from timestamp back to string.
HTH