TSA and 403(b)'s are the same money. 403(b) is the tax code that refers to a special investment vehicle for teachers, school employees and non-profit workers. Generally, if the money is invested in a fixed annuity, it is called a TSA. If it goes into an investment product (either a mutual fund or VA), it is generally refered to as a 403(b). Mutual funds are acceptable investments for this type of retirement plan.
Of course, the school district determines what annuity and mutual fund companies to make available to their employees. New companies can be added, but require at least 5 people in the district to sign up for the same company. Another limiting factor is that many mutual fund companies will not sign the school district's "hold harmless agreement" which vary from district to district, ranging from mild to rediculously legalistic.
As for annuities in a tax-deferred investment, it really depends on the individual case. Even as a commissioned advisor, I am not a big fan of using VA's in a tax-deferred retirement plan. While I'm well aware of the benefits, I don't find it to be a good fit for my client base. It can be a very good fit for the right investor.
Fixed annuities are a different story. The key benefit to a fixed annuity is not tax deferral, but the fact that it's "fixed". The client knows their money will be worth more tomorrow than it is today. This can be very comforting for more conservative investors. I can tell you the clients I signed up for a 5yr guaranteed rate of 7.15%, 3yrs ago, are thrilled with their performance. Fixed annuities can be an important addition to a diversified portfolio, but like VA's they aren't for everyone or every situation.