It's a simple question, but I'm having trouble finding an answer:
What's the purpose of having separate checking and savings accounts? What benefits does it provide?
For some people, it's easier to stick to a budget if they have separate checking and savings accounts because they can deposit funds directly into their savings account and not have those funds accessible by debit/credit card, checks, etc. This allows people to pay themselves first and accumulate savings, while making it slightly more difficult to spend those savings on a whim.
One a more technical/legal note, one key difference in the United States comes from Regulation D. §204.2(d)(2) of the law limits you to six withdrawals from savings and money market accounts. No such limit exists for checking accounts. Regulation D also forbids banks from paying interest on business checking accounts.
In the simplest case, checking accounts and savings accounts are a tradeoff between liquidity and return. Checking accounts are much more liquid, but won't necessarily earn interest, while savings accounts are less liquid because of the withdrawal limits, but earn interest.
Nowadays, however, sweep accounts blur this line somewhat because they function like checking accounts, in that you can write an unlimited number of checks, make an unlimited number of withdrawals, etc. but you can also earn interest on your account balance because some or all of the funds are "swept" into an investment account when not in use. The definition of "in use" can vary from business to business and bank to bank.
A checking account is instant access. It can be tapped via check or debit card.
A savings account is supposed to be used to accumulate cash for a goal that is is longer term or for an emergency.
Many people need to separate these funds into different accounts to be able to know if they are overspending or falling short on their savings.
In the United States the Federal Reserve also looks at these accounts differently. Money in a checking account generally can't be used to fund loans, money in a savings account can be used as a source of loans by the bank. An even greater percentage of funds in longer term accounts can be used to fund loans. This includes Certificates of Deposit, and retirement accounts.
If your debit card/ATM card is stolen or lost, someone else might be able to withdraw money from the checking account that it is tied to, or buy things with the card and have the money taken out of the checking account to pay the merchant. Subject to daily withdrawal limits imposed by your bank, a considerable amount of money could be lost in this way. At least in the US, debit or ATM cards, although they are often branded Mastercard or Visa, do not provide the same level of protection as credit cards for which the liability is limited to $50 until the card is reported as lost or stolen and $0 thereafter. Note also that the money in your savings account is safe, unless you have chosen an automatic overdraft protection feature that automatically transfers money from your savings account into the checking account to cover overdrafts. So that is another reason to keep most of your money in the savings account and only enough for immediately foreseeable needs in the checking account (and to think carefully before accepting automatic overdraft protection offers). These days, with mobile banking available via smartphones and the like, transferring money yourself from savings to checking account as needed might be a preferred way of doing things on the go (until the smartphone is stolen!)
Additionally, it used to be the case that savings accounts would have a noticeably higher interest than checking accounts (if the checking account paid any at all). So you would attempt to maximize your cash working for you by putting as much as you could into the savings account and then only transferring out what you needed to cover bills, etc into the checking account.