A stationary series has a stable mean, a stable variance, and an autocovariance that depends only on the gap between observations. Price levels fail on the first count: the mean of SPX in 2005 tells you nothing about 2025.
This matters because nearly every statistical tool assumes it. Regress one non-stationary price on another and you can get an R-squared of 0.9 between series that have nothing to do with each other, the classic spurious regression. Differencing to returns, or forming a cointegrated spread, restores the assumption.
Weak stationarity is enough in practice. Returns are not perfectly stationary either, because volatility regimes shift, but they are close enough that models built on them degrade gracefully rather than producing nonsense.
Related: unit-root, adf-test, cointegration, random-walk