Public schools typically do not offer 401(k) plans, instead providing 403(b) plans which are specifically designed for employees of tax-exempt organizations and government entities.
The primary difference between the two is that 401(k) plans are offered by for-profit corporations, and 403(b) plans are for employees in public schools and certain tax-exempt organizations. The choices in investment options may differ as well, with 401(k) offering more broad options.
The plans are similar. Both plans have identical base contribution limits and allow for tax-deferred growth, but 403(b) plans may include a "15-year rule" catch-up provision for long-tenured employees.
You can read more about the differences here: Tax Benefits of Retirement Accounts: Comparing 401(k)s, 403(b)s, and IRAs