Bandwidth is how wide the data pipe is between your service and the internet/users and is usually measured in bits per second [or megabits, or gigabits, etc]. Transfer is how much data you can transfer over a measured amount of time - usually a month.
So if you have 1 TB of Transfer it says nothing about your bandwidth. You could have a 1,000 Megabit connection, 100 Megabit, 10 megabit, or less - which is how much bandwidth you have.
These terms have become somewhat interchangeable in this industry but the reality is that they aren't - they're two different but related things.
If you have 1 TB of transfer that means you can transfer / upload / send 1 TB [ ~1,000 Gigabytes ] of data from the server to visitors/users.
If, for example, your website is 250 kilobytes on load - you can serve about 3,906,250 page loads per month [assuming no caching is in use on the client side / browser / anywhere else]. In reality you'll be able to serve more than that because you're surely going to have repeat visitors that have some of the data cached and, as such, they don't get sent the full 256 KB every time.
In reality - most sites on shared hosting use somewhere between 1 and 5 gigabytes of transfer/month. Most shared servers use somewhere between 5 and 50 megabit [with higher spikes, but the average is 5 to 50 or so].