iOS 6.1 Jailbreak Untethered. Reasons to Jailbreak?
I drafted a list of conceptual-level reasons once. The summary: you can make your device work better for you in a bunch of ways that Apple doesn’t allow.

• Add new features that Apple might never include in iOS: cool experimental interface improvements, extra features that require slightly increased battery or memory usage (but are worth it), and useful features they haven’t thought of yet. When independent developers have ideas for improving iOS, they don’t have to try to convince somebody at Apple to add them in the future – they can implement them and share them right away.
• Turn off things you don’t like. Instead of having to accept the default iOS behaviors that Apple estimates are best for you, add new options – improve, disable, or replace features according to your preferences. This includes fixing ways that iOS favors Apple’s default apps instead of independent App Store apps you might like better.
• Add new functionality to major parts of iOS where Apple doesn’t allow independent software – widgets for your lockscreen, Siri, and Notification Center.
• Customize your device to fit with your own style instead of Apple’s style, changing aesthetic elements that Apple would never let you modify.
• Look at your files. Apple believes that you shouldn’t see the filesystem; get access to your device’s filesystem and use it.
• Add insight. Become the administrator user of your own device and make your own privacy and security choices – get full insight into the software running on your device (including the ability to verify the claims Apple makes about it) and full access to change the software running on your device.
• Run apps you’ve written without paying Apple for the privilege.
• Add ways to remotely access your device from your computer or other devices.
• Add the option to restore your device to non-latest iOS versions by directly saving your device’s SHSH blobs for your current iOS version.
• Add new features (and sometimes security fixes) to older devices that no longer get official iOS updates.
• Add features that Apple can’t officially support due to their obligations to carriers.
• Choose to install software according to your country’s laws, not according to the laws that Apple is bound to (and the morals they’ve chosen).