The easy way is to backup the chats on the old phone, install WhatsApp on the S5, and restore from the backup. I'd fill in the obvious blanks if I'd followed this route. I didn't.
Long story short, I installed WhatsApp on my S5, which disabled WhatsApp on the old phone (DROID RAZR MAXX). The workaround for a broken WhatsApp on the old phone isn't for people without some sense of how Android file structure and file transfers work (explaining that is outside of the scope of this note). Anyway...
I used Bluetooth File Transfer because I have it on both phones.
- Uninstall WhatsApp and all data from the S5, or don't install it now
- Copy /storage/sdcard0/WhatsApp
- Paste the copied folder in the S5's /storage/emulated/0/
- Install WhatsApp on the S5 and validate WhatsApp with the correct phone number.
- WhatsApp will offer to restore old data. Do the restore.
- WhatsApp should now have all of the chats, etc. from the old phone
It ain't pretty, but it works.