A checking account:
– usually accrues no interest
– can used to pay bills via checks, debit card transactions, and ATM withdrawals
– usually has no limit on the number of transactions that can be done
A savings account:
– accrues interest (though not much these days)
– typically limits the kind and number of transactions that can be done with it
So basically a checking account is where money for your day to day payments flows through, while a savings account is where you keep your savings.
In the US, when you put money into a checking account, the bank has to keep 10% (unless it is small bank, then the number is smaller) in reserve, and it can lend out the rest to people seeking loans (as long as it meets other conditions not relevant here). If you put money into a savings account, the bank can lend out all of it without keeping any in reserve (as long as it meets other conditions not relevant here).
So, the bank can lend out more of the savings deposits than the checking deposits, which makes savings deposits worth more to them.
Because the regulators impose these differences in regulation between the two accounts, they don’t want banks to create something which is called a savings account but acts like a checking account. So they have regulation D, which forces banks to limit savings account, such as “6 transactions per month” limit.