How do APRS affect how much interest you pay?

Could you elaborate on how APY, or Annual Percentage Yield, impacts the total amount of interest that one pays on their investments or loans? I'm particularly interested in understanding the mathematical relationship between APY and the interest accrual process, and how variations in APY can significantly alter the final balance over time. How does it differ from simple interest calculations, and what factors should one consider when comparing APYs across different financial products?