Biodiesel is carbon neutral. But burning it still produces NOx and (probably) carcinogenic PM (particulate matter). So biodiesel is bad for the local climate/environment but not sure about global. The Europeans seem to be making great strides in PM traps for autos but even so, they are a long way off California's SULEV standards (which the Prius meets).
Alex, I don't know the difference for sure, but here are some guesses. Climate neutral would mean CO2 neutral, methane (CH4) neutral, CFC and HCFC neutral, NOx neutral, and habitat-destruction neutral. A tall order.
A good indicator might be to calculate the ecological footprint of the business. I'm sure that Mathis Wackernagel could help you with that. (If you contact him, please say hello for me.)
There are no official definitions of either "climate neutral" or "carbon neutral," so they're pretty much used interchangably. I'm sure there are some deep-dive technical differences, but they're lost on me . . . and pretty much everyone else who uses these terms.