Hashcash is the mining function used in bitcoin.
There are minor differences between the variant of the hashcash mining function used for X-Hashcash mail stamps and the one in bitcoin: a) hashash difficulty can only double or halve, bitcoin uses more fine grained difficulty adjustment; b) bitcoin is updated to use the SHA256 hash vs SHA1 hash in hashcash; c) bitcoin runs the hash twice for added security margin.
For comparability the bitcoin difficulty can be converted to hashcash (log2 expressed) bit difficulty by taking log2( bitcoin difficulty ) + 32. Hashcash stamps are 20bits, apr 2013 bitcoin was 55 bits and growing. Thats 2^35x bigger = 34 billion times.
Hashcash difficulty is static and eroded by Moore's law currently 20 bits.
Inflation vs deflation: the only current mechanism to upgrade hashcash default (20 bits) is via a sofware update though it can be overridden on the command line. Bitcoin on the otherhand has inflation control, with a dynamically adjusted difficulty aiming at a fixed rate of production. Inflation control is a major innovation of bitcoin over hashcash, and RPOW.
Note hashcash is just the mining function used by bitcoin. Bitcoin, which is the work of the psuedonymous Satoshi Nakamoto, is an extremely clever innovation and invention comprising multiple features and cutting edge new concepts, not fully realised in any previous electronic cash scheme. Bitcoin represents a leap forward in electronic cash technology demonstrating for the first time that a respendable, distributed, virtual scarcity based system could be built. There were earlier ideas that are similar however as far as I could gather Nakamoto was not aware of B-money, and presumably not bit-gold either has he does not reference that in his paper. The four main features of the bitcoin network are a public transaction ledger (in fact a transaction log because it exhibits cryptographically enforced append only properties), a p2p network for p2p transactions and distributed management of the security of the transaction log, a novel inflation controlled whole network mining difficulty allowing the creation of virtual scarce bitcoins, and finally smart contracts. Even without smart contracts thats a first. Smart contracts are icing on top, and also a first, with publicly auditable self-execuing smart contract.
Because of the byzantine threat models in maintaining a distributed transaction database, votes are placed on its correctness in proportion to the computational power of the clients. There is a reward for the computational power which is to create virtual, scarce, bitcoins at a fixed controlled rate, with difficulty of the work dynamically adjusted to keep the rate of production approximately fixed as the computational power of the network grows or shinks.