Greetings Alex,
You should be able to download the exact same app from the iTunes store to a different device at no extra cost. Its possible you might be downloading an iPad only version of an app you happen to have on your iPhone although I'm not familiar with an example of this.
If you are being charged:
1. Make sure you are using the same iTunes Store account (AppleID) on both devices. Your purchases are tied to the AppleID so if you don't use the same one to download the app that you used to buy it the first time, you'll be charged.
- On the device you originally bought the app in question: Go to Settings > Store -- note the AppleID here
- On the device you re-downloaded and were charged on: Go to Settings > Store -- note the AppleID here
If these AppleIDs are different then you have your answer. You are making purchases with two different accounts so logically you've been charged twice. Decide which AppleID you wish to use from now on and on the device which does not currently have that ID and tap on it in the Store section of the settings and login with the other AppleID.
If these AppleIDs are the same then something is going wrong in the process. Contact iTunes Store support email to have them take a peek: http://www.apple.com/support/itunes/
Hope that helps.