What is MojoSetup?
A cross-platform, standalone installer.
...With the Loki graphical installer (loki_setup) being fairly old, Ryan C. Gordon "Icculus" recent set out to work on version 2.0 of loki_setup. However, he came to the conclusion that he wanted to rewrite loki_setup from scratch, and thus MojoSetup was born. Ryan's MojoSetup is designed specifically to solve shortcomings in loki_setup while making this the next-generation graphical installer for Linux and other *NIX platforms. MojoSetup uses the Lua scripting language instead of XML, uses plugins for the user interface, a rollback mechanism for uninstallation, and the installer can access packages via FTP and HTTP (in addition to CDs, DVDs, SMBs, USB drives, etc), and is all distributed under the zlib license. The interface is also modern and Gordon had referred to it as looking like an Apple software installer...
This is the official site.
Which should I choose? What are the benefits of a .mojo.run file?
From what I see, it provides an easier way of Installing and Uninstalling software. Without going through the package manager. Because you see not very distribution has a Ubuntu Software Center, Synaptic, Gdebi or Muon Software Center for GUI.
How well integrated is it with Ubuntu?
It seems like it's a Universal solution and not geared specifically towards Ubuntu.
.mojo.run files? Should I choose one over a tarball?
Finally answering your question, mojo.run files are Graphical Installers. Should you choose one over the other?, do you prefer a nice graphical installer or the .sh file you'd run in the terminal?.