Bancor & Waves compete on some aspect of their use cases, and are potentially complementary so long as both ETH & Waves blockchains thrive.
 
Their competitive aspect lies in the fact that both Bancor & Waves aim to be the creation medium behind a whole slew of new tokens representing countless new types of value.
Bancor has two major differences to Waves here:
 
The first is that Bancor tokens are fully liquid from day one. So even if its some tiny token that never aims to become worth millions or be traded on a large scale... it's still always immediately exchangeable for any other coin or token on the entire Bancor network (including ETH and every major ERC20 token)
 
Waves has their decentralized exchange, of course, but it still requires that you find a second party to trade with, which rules out small-market-cap tokens.
 
The second is that Bancor is build on Ethereum tech. We don't compete with ETH at all, we are a use case of ETH-built technologies and coins. At the moment, that means we're backed by a more developed & successful blockchain, and therefore have a larger set of currencies and of users to launch with.
 
However should Waves succeed and become a widely used blockchain of tokens, the two can indirectly collaborate. Tokens on both Ethereum and Waves blockchains can be 1:1 tokenized for each other. Waves Token backed ERC20 tokens could use the Bancor network for liquidity & as reserve tokens.