Both Mountain Lion and Ubuntu 12.10 provide a fairly polished feel right out of the box. I also found that both operating systems offer an application store to locate and install new software. And both operating systems support cloud storage and notifications.
Where the two desktop experiences begin to divide is what is provided out of the box and what's not. Ubuntu offers LibreOffice, a fully functioning office suite. OS X offers Reminders, Notes, AirPlay, Calendar, plus a number of other applications. However, most of the applications each operating system offers out of the box can be downloaded for the other OS. For example, OS X users can easily install LibreOffice, and Ubuntu users can install reminder, note-taking, AirPlay and calendar applications easily from the Ubuntu app store. While there are some applications available for one operating system and not for the other, generally speaking there are comparable alternatives out there for both.
Where the two desktop experiences differ greatly is that OS X has the advantage of syncing with Apple-branded portable devices and Ubuntu currently does not. The up and coming Ubuntu Phone could potentially offer some relief for Ubuntu in this area, but currently it's a definite sore spot for new Ubuntu users. The reason why this matters is that support for these devices is assured over time. As Android device owners realize, the ability to mount one's Android device is hit and miss. Apple users rarely experience this as their Apple-branded devices "just work" without any hassles.
Despite Ubuntu's challenges in the areas listed above, I've found that Ubuntu has a clear advantage in the customization space—themes and alternative desktop environments. If Unity frustrates the end user, they can easily install an alternative desktop manager. For the less geeky, themes are also available that provide a higher degree of customization than what can be found in OS X.
Both OS X and Ubuntu offer a solid means of backing up their desktop. Ubuntu's use of DejaDup targets individual directories instead of true system-wise snapshots. It works best for backing up your home directory and leaving the rest of your system alone. Sadly, this doesn't backup your applications, rather only your application settings and other home directory contents. The contents of your Ubuntu backups can either be saved to the Ubuntu One cloud service or to a local storage device.
On the OS X front, iCloud performs a similar function. If you're wanting to back up to a local storage device under OS X, you would do so via Time Machine. Where OS X differs with its localized storage is that instead of merely storing directories locally with Time Machine, this backup tool offers incremental system "states" from which seamless restoration is possible.
Winner: Both operating systems offer tons of applications and an app store for downloading more. OS X wins by providing refined features such as dictation and parental controls. Ubuntu wins with desktop control and customization. Where things become fuzzy is with OS X's Time Machine. Out of the box, the Time Machine/iCloud combo wins over DejaDup. However, Ubuntu users could install a Time Machine-like program called Back In Time which offers rsync functionality and mirrors much of Time Machine's abilities. The Time Machine/iCloud combo wins from the end user's ease-of-use perspective.
When someone wants to install OS X (legally), they're doing so on a Mac or buying a new Mac with OS X pre-installed. On the other side of the arena, however, you can install Ubuntu on anything you wish. In addition, Ubuntu (Linux in general) goes out of its way to make this installation process as painless as possible by supporting a wide range of hardware choices. OS X is designed for Apple-branded hardware only. And while one can, in theory, install OS X on a PC, this isn't considered legal, and it's certainly not simple.
Ubuntu users will find that a quick search engine query will show off a number of solid Ubuntu-specific PC vendors who will support Ubuntu out of the box. You'll also find that there are far more hardware choices for Ubuntu pre-installed than there are Mac choices.
For diehard users of the two operating systems, this may not be a concern. After all, both platforms offer enough pre-installed choices to satisfy their individual audiences. But for anyone who values hardware selection over an operating system, this could be an issue.
Winner: With regard to pure hardware selection for pre-installed choices, Ubuntu buries OS X. Both operating systems are available on well-supported hardware. However, Ubuntu has the advantage of being able to support non-Ubuntu specific devices thanks to DVD and USB bootable media and licensing.
I could easily make an article series out of this, since this comparison barely scratches the surface of the differences between the two operating systems. I'd love an opportunity to dive headfirst into application titles, peripheral compatibility, and so forth. But instead, I'll close this article with the following thoughts.
Ubuntu wins with cost and overall value. OS X wins with refinement and tight integration. Ubuntu offers its users greater choice by providing FoSS and proprietary applications in their app store, while OS X limits their offerings to freeware and paid apps. The differences between Ubuntu and OS X range from vision down to individual user need.
Which operating system is the best? My advice is to read everything above carefully and determine what's important to you. Speaking for myself, Ubuntu (and other Linux distros) will continue to be where my time is spent. They offer me the experience that best meets my needs. Your own experiences may vary.