The actual problem of your crontab line is not the
$() or the backquotes. The problem is the percent sign
%. It has a special meaning in crontabs.
From the manpage:
Percent-signs (%) in the command, unless escaped with backslash (\),
will be changed into newline characters, and all data after the
first % will be sent to the command as standard input.
If you escape the percent sign with
\ it should work as expected:
* * * * * echo $(date +\%F) >> /tmp/date.txt
* * * * * echo `date +\%F` >> /tmp/date2.txt
both work on my site.