For this answer I am assuming that regulation D will be re-enabled when the current crisis is over.
Then the major difference between savings and checking for the consumer is the number of transactions that can be done in a month.
Yes both have FDIC coverage, as do Certificates of Deposits, and "money market" funds. FDIC coverage also extends to Roth or traditional funds deposited with the bank.
All of this is true for both banks, and credit unions.
Why do some banks pay interest on checking accounts?
Interest is the way they attract money into the bank. One way they also attract money is by offering other incentives such as free checking or a better interest rate if you setup direct deposit. They hope once you do this then you will keep making deposits and you will eventually have both a checking and savings account. They also may give a better rate on a CD or auto loan if you have large enough balances. They may also do this if you add a debit card to the mix.
The decision to offer interest on checking is dependent on how the flow of money is performing. Once the funds are in the bank they offer better interest rates to get customers to move it to savings, or a CD. Once it is in one of those accounts, then it can be a source for loans which is how the bank generates the funds to pay the interest on deposits, pay for their employees and other expenses, and leaves them with a profit.
Decades ago when my credit union paid 6% for a money market account, and 3% for a savings account, they only paid 1% for checking. That worked for them. Car loans, student loans and mortgages were charged rates above 6%.
Now when a savings account earns less than 0.5% and an auto loan or mortgage is at 2%, then there isn't much room for checking accounts to earn interest. Unless of course the bank needs to get an influx of new money, so they offer a great rate on checking account, so they can later convert you to other products...