I am replacing an old electric range (240v) to a newer electric range (208v). Will I need new wiring, a bigger breaker, different outlet or all the above? Or should it work with the same wiring as the older unit?

What does normenclature tag on new stove say.
Assuming this is a residance in US you don't have 208 voltage. Wire sizing and breakers are amperage dependent, not voltage.

New stove is probably 240/208v.

Where are you getting the 208 VAC, this is 3 phase in the US. Where are you located? Why use a 208VAC range?

Assuming your home currently is 240 volt,s then the new range needs to be rated 240 volts.

208 Volt appliances are for those large multi dwelling homes, apartment, and condominiums that are served by a 120/208 volt three phase Main service, and each dwelling is single phase.