If you have a tablet or a smartphone, you have two basic choices. You can get the Overdrive app or the Kindle app. Which one you want depends on what you need it for.
Do you want to listen to audiobooks? If so, then rule the Kindle app out. It's strictly for e-books, so go with the Overdrive app (aka Overdrive Media Console).
If you're reading just e-books, then read on.
Do you have multiple libraries you're using? If so, go with the Overdrive app. You can add multiple libraries to the "My Library" list. If one library doesn't have the new book you want, the other one might.
Do you already have the Kindle app and don't want to install another app? Then stick with the Kindle app. You'll have to use your browser (Chrome, IE etc) to get to the library's website to get the books, but you won't have to install another app.
Do you want one-stop-shopping? Then go with the Overdrive app. You pick what library you want to get your new book from, and it opens up right in Overdrive. Everything, actually, is done from the Overdrive app once you get it downloaded.
The one drawback (in my mind) of the Overdrive app is that to use it the first time (to get that first book), you have to get an Adobe ID. It's free, so don't worry about it charging your credit card. But it's your e-mail address and a password. You can register for one right on the Adobe website.
As for ease of use, I find the Overdrive app easier to use and more convenient. The Kindle app is nice if I don't want to get another app, but there are a few more steps since it takes you to the Amazon store to check out the books (a marketing way of trying to get you to buy it), but it's not too bad to use once you get used to it.