Attempt to download app, says it's already installed on my Iphone4
There is a difference between an app that has been purchased and one that is installed. To install a purchased app, open the App Store app on your iOS device.

Click on the Updates button in the lower right corner to advance to the Updates pane:



Click in the Purchased area near the top of the screen to advance to the Purchased pane:



Select the Not on this iDevice tab to see purchased apps that aren't currently installed:



Select the app you're looking for from the list and install it.