Recently I saw this on bitcoin.org and just downloaded it, what I noticed it's bitcoin address is changing from time to time. Is this a safe wallet like others?
As far as web wallets go, I would say that Greenaddress is safe, yes. They are owned by Blockstream and have a very good reputation. I've used them in the past and never had any problems. It should be noted that they use 2-of-2 multisig addresses, which create larger transactions and therefore have higher fees. This has been somewhat mitigated since they implemented Segwit.
Your default receiving address will change after you have received a BTC payment. Other services usually do the same.
And it's mnemonic is like it's private key?
Not exactly. Private keys correspond to key pairs. So, each Bitcoin address has a corresponding private key. Your mnemonic is a "seed" that gives access to all the private keys in your wallet.
Note: Unless things have recently changed, your money is stored in 2-of-2 multisig addresses. Your seed provides one of the keys. In case you need to recover your wallet, Greenaddress uses the 2nd key to pre-sign an nlocktime transaction that spends the money back to an address under your control after a certain time. See more here:
https://greenaddress.it/faq/
