Does Electrum use Bitcore Apis as it's backend or not?
If not, then what is the purpose of developing Electrum when Bitcore Apis provides all the features Electrum has (like HD wallets and master seeding)?
In fact, electrum actually predates bitcore by about 2 years.
As to why there are multiple implementations of similar APIs and softwares, the answer is really "Why node?". Even the bitcoin node software has multiple implementations (Bitcoin Core and btcd, for instance).
This is a very open source friendly space, and it is generally actively encouraged to have multiple approaches to a problem.
Electrum uses an Electrum Server as an API, which holds information on block headers, UTXOs and information on address balances and historical transactions. Electrum does all key management on the client and no critical user information are stored on the server.
Bitpay (Bitcore wallet) uses a Bitcore Wallet Service (that does wallet creation and key management) and that in turn uses the Insight API (which stores block / transaction data). Insight communicates with Bitcore which sits on top of Bitcoin.