If you use the 'Forwarders' menu of your cPanel to forward email from email@example.com to firstname.lastname@example.org, all email sent to email@example.com will not just be forwarded to firstname.lastname@example.org, but will also be saved in the mailbox of email@example.com (unless you haven't created an email account for firstname.lastname@example.org in the 'Email Accounts' section of your cPanel).
This could be a problem if, for example, you have little free disk space available on your account.
To prevent email forwarded to email@example.com from being saved in the mailbox of firstname.lastname@example.org, do not use the Forwarders menu. Use the User Level Filtering > Manage Filters > Create a new Filter menu of your cPanel instead. Create a filter to forward and then discard email, like so:
From matches regex
Redirect to email email@example.com (hit the '+'-button here)
The above will affect all mail sent to firstname.lastname@example.org. If you only want to forward a specific subset of mail sent to email@example.com, change the 'Rules' accordingly.
Note: while simply deleting your firstname.lastname@example.org email account can be a solution to this problem in some cases, in other cases it is preferable to keep an email account for email@example.com. An example of the latter: when you want to reply to mail forwarded from firstname.lastname@example.org to email@example.com with your firstname.lastname@example.org address, it is a good idea to send it through the SMTP server of email@example.com (in order for the reply to not appear as a redirect to the recipient, as would happen in Outlook clients etc.). To use the SMTP server of firstname.lastname@example.org, you need to have an email account for it. This option is also available for Gmail users.