The anonymity of bitcoin transactions has been a challenge for financial regulation. But it has also been a challenge for those users who wish to hide their identity.
Using the same Bitcoin address: not recommended
The only reason to provide a different bitcoin address for each transaction is to protect your anonymity. If you reuse the same address, your entire transaction history can be instantly available to anyone simply by logging on to blockchain.info Your transaction can be traced and anyone can calculate how much money you have and how you spend it. On the contrary, with the use of a different address, the bitcoin transactions are hard to trace and it becomes almost impossible to prove the identity of the user, i.e. that the owner of transaction A is also the owner of transaction B. It requires someone to constantly monitor your page and know all the different addresses you are using to be able to trace your transactions and break your anonymity.
The RNG factor
The Random Number Generator (RNG) is extremely important in protecting your wallet. If it is not broken, as it happened lately with the Android, it can generate really complicated equations, which make it extremely hard for someone to break your server and find out about your transactions. However, by reusing the same bitcoin address, you potentially reveal a traceable system of equations, making possible for someone to break your private key.
Store multiple bitcoin addresses in your wallet
You can save multiple bitcoin addresses in your wallet and use a different one for each transaction. The good news is that addresses do not expire, they are valid incessantly and, therefore you can use them once to protect the anonymity of your transactions. In addition, given that the namespace is beyond human comprehension, theoretically you can generate millions of addresses and perform your transactions anonymously. However, it is equally important to secure your wallet in order to protect your private from being violated.