Bootstrap, jQuery Mobile and PhoneGap all solve different problems.
Bootstrap is a responsive web framework. Its goal, like all responsive web frameworks is to adapt your view layer to the device doing the viewing. Giving you different presentations for mobile phones, tablets, PCs, etc.
jQueryMobile is a mobile web application framework that aims to provide a cross-platform mobile application written entirely using JavaScript/HTML/CSS.
PhoneGap is essentially a wrapper (browser) around an existing web application that lets you access native device APIs (like camera, bluetooth, etc.) using JavaScript.
From my own personal experience, which is by no means a comprehensive guide:
Bootstrap
- Pro: Easy to work with, easy to customize.
- Pro: Lots of people use it.
- Con: Lots of people use it, so your application may look like all the rest.
jQuery Mobile
- Pro: A nice mobile-friendly UI.
- Pro: Easy to get started, especially if you're familiar with jQuery.
- Con: The more data injection you need to do, the more complex your app becomes.
- Con: When I last used it, 1.0, it was fairly buggy across devices.
PhoneGap
- Pro: Easily lets you build cross-platform web apps with JavaScript/HTML/CSS and have access to device APIs
- Con: When I last used it, this is mostly for Android, the webview it used was brutally slow.
For trivial applications, my recommendation is to definitely go with cross-platform development. It'll be relatively quick and easy to deploy. Having said that, once you start increasing application complexity, my experience has been to always go native, or at the very least some hybrid solution. Native applications just have a better "feel" at this point in time and more importantly, going native lets you build applications within the design of the platform. Cross-platform is nice, but most Android users don't want their apps to look like iOS apps and vice versa. While it may take longer to build native applications for multiple platforms, I think you ultimately deliver a better experience to your users.
Just my two cents.