Why is the content of a Receipt for New Memberships different to that for a Renewed Membership
The receipt templates are at Mailings > Message Templates > System Workflow Messages

Under Memberships, there are two receipt templates for on-line and off-line.  From a quick look at those, I'm not clear how your problem arises.

What version are you on?  Templates do get changed so your problem may already have been fixed in later versions.

Are you using the default versions?  If not, it will show you a 'Revert to Default' and 'View Default' menu link as well as 'Edit'.  If changes were made to your templates a while ago then they may be very different to the current default version.  In this case, you need to compare your customised version with the default version.

If you are on the default version then you can easily experiment with some changes and use the 'Revert to Default' if you mess up.  Otherwise, you can save a copy somewhere, try changes and copy back your original saved version.

Hopefully that gives you enough to get started and narrow down the issue to a specific template and area of the file.  If you update your question with more detail someone can help you further.