Hi....
My thoughts on this (someone please correct me if I am in error)...
If it's a standalone
.deb package you're downloading from a website, then, theoretically, there is across-the-board compatibility among the Debian based distributions of Linux. However, you can easily get into dependency, version and breakage issues pretty quickly.
It's safer to install software from your package manager but if you want to install something from the web, a good way of approaching this might be this...
1. Which repositories is your copy of Elementary downloading from, Ubuntu or Debian?
2. After you've determined that, what version is being used (i.e, Trusty, Jessie, Sid, etc.?)
That will give you an idea which distribution and version to download from. Hopefully, you will experience less problems this way. One way to find out what repositories your OS is downloading from is to have a look at your "sources.list" file, found under "etc/apt" in your file manager.
It's certainly possible to install packages from either of the two distributions, regardless of version, I've done before (and had it work,) but just be
very careful.
Clever username, by the way.
Regards...