Short answer - banks make loans, and obviously some of these will go wrong. This risk is handled in two separate ways (that are distinct; but obviously not independent of each other). If you had to sum up the distinction in single-sentence bullet points:
Provisions try to pre-book expected/likely losses on the current loan book.
Capital Adequacy (CAR) tries to ensure that the bank remains sufficiently well funded to continue to make loans.
So I run a bank with 100 loans (that are my assets). A is mortgages, B is credit cards, C is student debt, D is auto loans etc. etc. For each category, I have to book a provision against every loan I make (that is set by the perceived riskiness of that type of loan). For every loan payment that is overdue, I have to increase the provision a bit against that loan (missing a mortgage payment is a sign of trouble, but it's not game over). And for every loan that is overdue by months, it becomes a "bad debt", heavily provisioned against a likely expected recovery rate (high for mortgages, bad for credit cards etc.)
This provision is a simple accounting entry: a balance sheet debit/liability and an income statement credit/cost. When my borrower misses a payment, this generates a cost and a liability, that reduces my equity. And when he covers this a week later and back on track, the reduction in provisions is a profit and a reduction in liabilities that increases my equity.
It's (mostly) as simple as that. The only point to add is that sometimes bank profits can be swung as much by the provisions (as expected losses) as by actual losses. Coming out of the GFC was a case in point. Banks didn't find new customers eager to lever up again... but they did find that the absence of a true Great Depression 2.0 meant that the risks weren't as dreadful as feared at the worst. Unlocking those provisions restored a lot of value.
Capital adequacy is similar, but different. The regulators want to make sure that a bank is sufficiently well-funded to be able to continue to make loans. Every time any bank makes a loan, it creates both the asset (the customer owes the bank) and the liability (the customer has a cash deposit). Theoretically, there is no constraint over how much any bank could lend to anyone. So the regulators demand that banks set aside a cushion of capital to cover the risks.
Unlike provisions, this does not change the balance sheet or move the P&L. Very simply, if I have 100 loans of mix above, I need to deposit X at my regulator. I create new loans; X grows. I take a hit and have to write off a loan, I better have sufficient funds above and beyond X to cover that eventuality. So the CAR essentially keeps me honest about my ability to make new loans.
Scenario - I'm too aggressive. I have 90 loans plus 10 govvie bonds as assets, with 90 credits and 10 equity on the liability side of my balance sheet. My bonds are parked at the central bank for 10% (of assets) CAR purposes. I want to make another 10 loans... except then, I would have 110 assets = 11 CAR; so I can't. But if I was running 80 loans and 20 govvies, I could easily add another 10 loans with my 10 equity.
CAR simply acts a brake on banks getting too leveraged and lending too much (yes, I know, I was there in the 00s and 2008 too ;-)
Where these two parallel systems (one risk and accounting, one capital and regulatory) interact is when provisions change. The profit or loss impact of that will increase or decrease equity. And this will improve or deteriorate capital adequacy in tow.
So they are related, even if one is simply an estimate of future losses on current lending; while the other is simply a current measure of the riskiness of continued lending in the future.
hope that makes sense.