The main difference is the type of employers who can offer them. Unlike 401(k) plans which are offered by for-profit companies, 403(b) plans are only available to employees of tax-exempt organizations. These are usually either schools, hospitals or religious groups. The names simply refer to the section of the tax code that outlines these plans.
For the most part, the two types of plans work the same way. While 403(b) plans historically offered more limited investment choices than corporate plans, they’ve recently begun offering a broader array of investment options. And while 401(k)s frequently have vesting schedules spread out over a few years, many 403(b)s vest immediately, or over a shorter period of time than in their cousins in the for-profit world.