The TI-85 was the second graphing calculator made by Texas Instruments. (The first was the TI-81). The TI-84 is a much newer version and has more functionality and memory. The only real advantage that the TI-85 has over the TI-84 is that it costs less.
As far as where you are allowed to use the calculator, almost every test or class that allows a TI-85 will allow a TI-84 as well. The TI-84 is a much better investment. (If the TI-84 is too expensive, look into getting a TI-83. This model is also newer than the TI-85.)