Uniswap is an automated market maker (AMM) that runs fully on-chain whereas AirSwap is a request-for-quote network (RFQ) that runs as a combination of off-chain (peer-to-peer negotiation) and on-chain (settlement by atomic swap).
There are benefits to each. With AirSwap there is no price slippage and trades are unlimited size. The trade-off is that it's easier for everyday users to provide liquidity to Uniswap. However, a successful outcome for RFQ is a few high quality liquidity providers covering a variety of digital assets, so AirSwap does not require the same level of democratization as Uniswap for the network to be liquid.