I have both savings and checking account with Discover as well as their credit card.
1) The cash back checking is a great way to earn cash back for those people who use a debit card. You will earn 10 cents for each debit card swipe and check clearance. Not much, but no other banks in my knowledge offers it. The debit card is a Discover debit card which is not as widely accepted as visa or MasterCard but most merchants now accept it. I personally rarely use a debit card for a transaction as credit cards have much better fraud protection than any debit card.
2)Their ATM network is big. You can withdraw money from any CVS ATMs and other ATMs also. One problem with these online banks is lack of depositing money. Like chase ATMs where you can deposit money, you can't do it with Discover bank. Even though I use mostly credit card and my salary directly comes to my bank account but sometimes my roommate pays me cash and I can't deposit that in the Discover Bank. (You can mail them, it's a risk from my point of view). So, if you are dealing with cash you must an account at a local bank. I deposit money into my chase account then ACH to discover.
3) Few people complained about their ACH speed. Until recently, if an ACH request I put in Discover to withdraw money from my Chase it took 4 business days to happen. Now they reduced it to 2 days. Alternatively, you can always ACH from Chase to Discover which took 2 days. Also, you can deposit a check through the app which hassle-free.
4) My Discover checking is my default payment account for every credit card, utility companies as I will never close it as it does not have any maintenance fee. I Chase checking I need to maintain 1500$ to avoid fee which is not possible for me to do always.
5) Their saving account has 1.15% APY just behind market leader CIT bank which offers 1.3% APY. So, I save all my money is the Savings account. Just because they are not traditional brick and mortar bank their operational cost is less so they can offer higher interest rate compare to chase which is 0.01%.
6) About the smartphone APP. The APP is great, comparable with Chase, fur better than the Bank of America in my experience. I think a local bank account(big bank or credit union) and Discover makes a great pair.
7) Discover Bank is not a member of Zelle pay. If it ever becomes a member you can transfer from your other Zelle accepted banks in real time. (https://www.zellepay.com/partners)