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In summary, you wouldn't use jQM to write a desktop web app, and neither is jUI appropriate for mobile web app. There's not much conflict to talk about.
It all depends on what you're trying to accomplish. If you need or want a different look and feel for your application on desktop and mobile platforms then you probably wouldn't want to use only jQuery Mobile. You'd be stuck with providing two versions of your application (one for jQuery Mobile and a second for jQuery UI on desktop as an example) and the support/maintenance involved with separate solutions. That is, unless you can figure out a way to make the two frameworks together which is a known challenge/issue.jQuery Mobile has broad support for the vast majority of all modern desktop, smartphone, tablet, and e-reader platforms. In addition, feature phones and older browsers are supported because of our progressive enhancement approach. We're very proud of our commitment to universal accessibility through our broad support for all popular platforms.