Difference between "reg save" and "reg export"?

The generated file is the difference.

save saves the branch into the Registry's native "hive" format. This preserves key ownership and ACLs, so is best suited for backups and restoring on the same machine.

export exports it into a textual format, suitable for distribution since the .reg format does not preserve any metadata.

It's not just that the generated files are different. I noticed that what happens when you use the .reg file vs when you use the .hiv file is quite different

When you import a .reg file, you don't have to specify a path. It writes to your registry based on the .reg file, and leaves new registry keys and entries created between export and import untouched.

Whereas when you import a .hiv file, you have to specify a path. New registry keys and entries created between export and import are deleted.

Here's an example to illustrate.

1. Create a dummy key with a dummy entry

   reg add hkcu\test
   reg add hkcu\test /v entry /t REG_DWORD /d 1
   

2. If you run reg query hkcu\test you will see this

   HKEY_CURRENT_USER\test
    (Default) REG_SZ
    entry REG_DWORD 0x1

3. Export to a .reg file and to a .hiv file

   reg export hkcu\test test.reg
   reg save hkcu\test test.hiv
   

4. Then create a new entry and a new subkey

   reg add hkcu\test /v new_entry /t REG_SZ /d "ayy lmao"
   reg add hkcu\test\new_key
   

5. If you run reg query hkcu\test you will see this

   HKEY_CURRENT_USER\test
    (Default) REG_SZ
    entry REG_DWORD 0x1
    new_entry REG_SZ ayy lmao

   HKEY_CURRENT_USER\test\new_key

6. Import the .reg file we created earlier

   reg import test.reg 
   

   If you run reg query hkcu\test, you will see the same output from #5.

7. Import the .hiv file we create earlier

   reg restore hkcu\test test.hiv
   

   If you run reg query hkcu\test, you will find that the new items we created in #4 is gone and we get the same output as #2.