Did you go to General > Usage > Storage? It will show you a list of apps and how much space each one takes up. Click an app from the list to delete it and free space. I also had to do this to make room for iOS7.
How much stotage do your iPhones have? (8, 16, 32, 64). You must bear in mind that storage capacity is finite and the more data (photos, videos, music, txt messages, emails, voice memos, etc) you store the less you have available for other uses, including software updates.
A Full iOS 7.1.1 download is a not insignificant 1.26GB, whereas an update from 7.1 to 7.1.1 is about 1/6th of that.
Mine is 8GB and I had to clear a lot to install iOS 7 but after it installed it had actually used very little of that memory and I just re-downloaded what I had deleted.
If you update your iPhone through iTunes, you don't have to delete anything!! It is only the over the air (OTA) updates that require space on a device as the device needs plenty of room to move files around. Using a computer obviates this need...