Yes, the VFD will improve the displacement Power Factor and if you are charged significantly for displacement power factor, there will be a cost reduction, but if you are charged for true power factor, then there may in fact be an increase in cost.
You need to determine a) what is being paid for and how it is being measured.
If there is a simple power factor penalty where the power factor is derived from true KW / True KVA, then the penalty may increase. Most likely it may fall a little, but not completely.
The addition of a VFD WILL NOT reduce the KWHr charge unless there are mechanical inefficiencies that can be improved by slowing the machine down. If the motor runs at line frequency from the VFD rather than from the supply, then you will actually use more KWHr due to inefficiency in the VFD.
Kens : yes, it is totally dependent on the type of metering used and how it is configured, plus how the tariff is structured.
The trend towards is resulting in a trend towards true power factor. Many power bills I have seen use KW/KVA rather than KVAR as the power factor indicator and as such, the distortion of the current waveform begins to become significant on the power bill.