Do alternatives to the default Twitter app use less data? I'm currently using the default Twitter app, and it's apparently used 13.9 GB since I purchased my phone in December. Do alternatives to the default Twitter app use less data, and if so, what are they?

Ideally, the app should not have a purchase price, but that isn't a hard requirement. I'm only using one Twitter account, not multiple accounts. I have an iPhone 5S.
I'm not aware of another app that uses less data but you can reduce your data usage on Twitter for iOS 10: 

Settings -> Data -> Video auto play -> Use wifi only

This will stop videos from playing automatically. I suggest you reset your cellular statistics and see how much data you use now. If you want to go even further you could turn off Image previews on that same "Data" page.