Difference between a checking and savings account

Limitations on savings accounts are now at the discretion of the bank.
Before anyone freaks out on my answer, read this article about changes to Regulation D relaxing limitations on withdrawals from savings accounts:

https://www.forbes.com/advisor/banking/regulation-d-has-changed-what-this-means-to-savers/

That being said, many banks may give you a certain number of transactions for free and then charge you a fee for anything beyond that, but this is a case-by-case situation.

Depending (again) on your bank, you may or may NOT receive interest on your checking account, and many times (but not always!) banks will pay this interest on a checking account ONLY if you maintain an average daily balance above a certain amount.

Banks offer interest as incentive to leave your money with them, and checking accounts are the most volatile (meaning they have the most frequent changes to balances as people pay bills and receive paychecks), so the only way to pay interest on checking balances (from a practical standpoint) is to set a minimum balance limit at which such payments are made.

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...

A financial institution will offer a number of distinct PRODUCTS. A savings account and a checking account are merely types of deposit products. They are governed by terms defined by the bank as the banker designs the product to attract depositors' funds. They do this in a highly regulated environment within many constraints defined by legislation, regulatory body edicts, and case law. The bank may represent things as regulatory limitations when it's simply not in the bank's business-interest. Or the reg might be there to protect the bank from itself. It depends.

When deciding between a savings account and a checking account, read the disclosures for the two instruments noting the terms of the accounts (it's a contract between depositor and bank) and comparing them. Pick the one that more closely suits your needs. You can't just assume a checking account is X and a savings account is Y. The details are something you need to consider.