Both are implementations of the Bitcoin protocol, however they are customized to make it easier to develop with. They are able to do this by storing all transaction and blockchain data in a database for quick queries, and along with this comes large disk space requirements (200-300GB).

Bitcore (I have used)
- NodeJS
- 200+GB required for storage as it holds all transaction data.
- Bitcore also has a library-only version (not a node), and a public API.
- Add-ons available (Block explorer API, UI & and Wallet Service)

Toshi (I have not used)
- Ruby
- 300+GB required for storage as it holds all transaction data.
- Toshi has a public API. As far as I can tell it does not have a library-only version like bitcore.
- A block explorer API and UI available. No wallet service, that has to go through coinbase which requires an application, credit check, etc.

As for your question about performance I can't really say since I haven't used Toshi, but Toshi's Github hasn't been worked on in over a year with only 200 commits. Bitcore is constantly being worked on with over 1,600 commits some as recent as 6 days ago.

Also I'd say outside usage would be a good guide as to which one performs better. I'm not aware of any well-known service using Toshi.

Localbitcoins uses Bitcore for their explorer. https://www.localbitcoinschain.com/
Counterparty uses Bitcore http://counterparty.io/news/counterparty-uses-bitcores-insight/
SatoshiLabs(TREZOR) uses Bitcore http://themerkle.com/satoshilabs-unveils-mytrezor-beta-and-switches-to-bitcore/