With a blockchain, a client downloads all the blocks and runs a local query to see if there has been any double spending of a coin (roughly). Here, a webcash wallet queries the webcash server for double spending protection instead. It asks the server, "has this webcash been spent already?". This is substantially faster, and it requires a different trust model from the regime of the trustless decentralized blockchain.

With regards to mining, actually right now CPU mining still works. When the FPGA and ASIC miners come, it would be (comparably) trivial to switch out the mining algorithm that the server recognizes.

If server trust is required, why not just put a database into it and store signed records like every bank does?

Yes, the server is a wrapper around a database. That's the point of this project. :)

That’s what this is.