Solution :
I know about the frameworks so I am trying to help you.I have already used both of them and each of them have peculiar conditions where one is better suited than the other one.
If you ask me then you should use the JQueryMobile for below conditions:
1. If you are looking for quick and very light weight mobile implementation of the web application.
2. If you want to learn it quickly JQueryMobile is certainly more faster to learn and implement than the Sencha Touch.
3. And the local look and feel is not your requirement.
4. Also while making the extension to the existing web application in which you have the relatively simple functionalities need to be extended to the mobile I found the JQueryMobile is extremely useful and very straight forward.
If you ask me then one should use Sencha Touch for below conditions:
1. If you are looking for the native look and feel.
2. If you need the higher degree of functionality on your mobile side like access to the native api's through your phone gap.
3. And if you are targeting some of the latest smart phones as its performance is very impressive.
