Note: This answer is restricted to Ubuntu/Debian.

In general, no, they do not mean the same. A "package" is like an installer; it's a DEB file, much like MSI files in Windows.

Unless the application is relatively small, it will usually be assembled from multiple packages, with each package containing one (sometimes more) components of the program. These components can be binaries (executables), libraries, help files (man pages) and even source code.

Conversely, some packages will install multiple small programs (binaries), and others will "collect" multiple related programs under one umbrella for ease of installation (meta-packages), but these are the exception rather than the rule.

In short: since most programs are made up from more than one package, they are not the same. :-)