In short, Linux Mint does uses exactly the same base as Ubuntu. In fact, if you check the contents of
/etc/apt/sources.list, you will see that Mint uses the Ubuntu archives for most of its software.
Most likely you have made an error in installing the software. Particularly when you say that you 'downloaded' the software, it implies that you didn't use
apt-get or another package manager to install the software. You should always attempt to install software in this manner as each package in the Ubuntu repository has been configured to integrate with the system (be it Ubuntu or Linux Mint), attempting to install from another source may be problematic.
The difference between Linux Mint and Ubuntu lie mainly in the packages which are installed by default, the artwork which is included and the installer itself.
Linux Mint is geared toward providing a clean and easy to use interfaces for new Linux users. The Mint developers have created a number of programs/packages to that end. This includes the Cinnamon and Mate desktop environments (originally forked from parts of GNOME2/3). Other things which have been included are desktop applications such as the Mint update manager and a favourite of mine Simple Scan.
However, as all these applications are Free Software, they are widely available outside of Linux Mint. In fact the Mint desktop environments along with other Mint software are available in the Debian repositories (this is where I get Simple Scan from). As Ubuntu is based on Debian, they are also available in the Ubuntu repositories.
Since this software is available in the Ubuntu repositories (and the Mint developers do not engage in the mass packaging of software required to maintain the own independent repositories unlike Debian/Ubuntu), Linux Mint uses the Ubuntu repositories as the source of most packages installed on the system. Although Linux Mint do provide their own repositories for new releases of their own software which may still be unstable or otherwise hasn't made it into Ubuntu yet.
Most of the Linux Mint developed software is actually installed from Mint repositories which take precedence over those in the Ubuntu ones. The packages in these repositories will either have a specific Linux Mint configuration or depend on another package which does. Also in the Mint repositories are packages containing the artwork unique to the Linux Mint distribution. However the number of these specific packages is tiny compared to the tens of thousands of packages available in Ubuntu. Package lists for the Linux Mint 16 repositories can be found here and here.