How is % special in crontab?
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.txtor
* * * * * echo `date +\%F` >> /tmp/date2.txtboth work on my site.