How transaction fees differ between different wallets? In particular, I'm wondering what is the difference (if any) of transaction fees between Bitcoine-core, MultiBit and blockchain.info?
Bitcoin-Core is a full node and can therefore build up a mempool and estimate fees. This is probably the most secure way how one can estimate fees. You can verify the source code and make your own improvements. The fee estimator allows you to estimate the fees for different confirmations targets (2-25 blocks).

MultiBit does either use a web-api and/or static fallback fees. You depend on third-parties, estimating fees for you. All SPV wallets cannot reasonable estimate fees.

Blockchain.info run its own full-nodes and use it's own fee estimation logic. You need to trust the estimations produced by blockchain.infos logic.