The difference is the size of the drop. A correction is a decline of 10% or more in a stock, index or market from its recent peak, while a bear market is a deeper slump, generally a fall of 20% or more in a broad index such as the S&P 500, usually lasting at least two months and accompanied by widespread pessimism.
Corrections are common and often brief: a CNBC and Goldman Sachs analysis cited by Investopedia found the average S&P 500 correction lasted about four months with a drop of around 13%. Many never become bear markets; of the 19 S&P 500 corrections between 1980 and 2020, five turned into bear markets.
Sources: Investopedia: Correction; Investor.gov (SEC): Bear Market