For an excellent full review about SIMPLY TWEET read Helen Crozier’s post http://www.myfabulousiphone.com/2009/09/iphone-twitter-simply-tweets-the-one-for-now.html
I started using the app and found it great to use and found the PUSH notification quite a novelty which I will use when I want to be notified immediately of replies or DMs. When I wanted to turn it off it was not very clear how to.
So here it is simply
1. Go to the iphone settings (outside of the SIMPLY TWEET app)
2. Select NOTIFICATIONS which is near the top of the list in settings
3. Once in the NOTIFICATIONS area select SIMPLY TWEET
4. Move all 3 toggles below to the OFF position to completely turn off the PUSH function