Why would anyone choose 403b over 457b?

403(b) plans are similar to 401(k) plans and are restricted to non-commercial employers such as elementary and secondary schools, colleges and universities, non-profit hospitals, and the like (charities?). 457 plans are a separate retirement program offered by States to State employees. I doubt very much that 457 plans have no early withdrawal penalty, but perhaps his State is different from mine which not only had an early withdrawal penalty but also poorer terms for withdrawal after retiring; it was better to roll over the whole amount into an IRA upon retirement.

If the OP has access to both a 403(b) plan and a 457 plan, the OP presumably works for a State College or State University or State Hospital (or perhaps a State School, though typically, States don't run schools: they are controlled locally). Generally, 403(b) plans are handled by HR departments at the institution (so signing up, making changes etc is relatively easy), while 457 plans are run through a State website through State entities called Central Management Services or something similar (and thus are a little more complicated and impersonal to deal with). Those who have access to both a 403(b) plan as well as a 457 plan can choose to contribute to one or the other or to both. Typically, the plans have different sets of investment choices, often don't have employer matching contributions as with 401(k) plans (or maybe the employer contribution is set to 0%), and also have separate maximum allowable employee annual contributions, so that it is possible to put away $38K+ (twice as much as with a 403(b) plan) towards retirement if one has enough salary in each paycheck to support the contributions.