If you flashed the original image on the 3B, you may recall that it was exactly the same as the image for all (most) of the other RPis; see the software download page. Therefore, it seems unlikely that the issue is incompatible firmware - unless it's a very old image; i.e. one that was released before the 3B+ was released.
After installation, it's likely that you made changes to the 3B's configuration that could render a backup image different enough to cause network issues on a 3B+ machine. Of course, there are many other possible explanations for the behavior you're seeing on your 3B+. Yes, I know that's not very helpful, but your question lacks detail that might provide clues. That's not a criticism - there are innumerable "details"! That said, a significant omission in your question is that you failed to identify the OS version; i.e.
If I were in your situation, this is what I'd do:
Set aside the 3B image for now, but don't lose it!
Flash your 3B+ with a fresh image - d/l from the "software page" I linked above.
Configure the 3B+ such that it works as you need it on your network.
Once you have your 3B+ up and running, you can
mount the image file from your 3B. This will allow you to simply copy any files you need from your 3B to your 3B+ . NOTE: Be aware that configuration files may change substantially between releases, but IIRC
/etc/dhcpcd.conf has been stable for several years now.
If you have trouble performing the steps listed, you will need to post those as new questions. Do not try to turn this into a long chain of comments with questions soliciting details for "how do I do this step?"! It's better for all of us to make the Q's and A's here as succinct as possible.