If you are happy with your current broker, always give him/her a chance to compete with commission rates, etc. It is better for both you and the broker in the long run. Your biggest benefit is managing a single account for things like margin requirements, etc. The broker's obvious advantage is that it less expensive for him/her give up a bit on commission than it is to get a new client.
That being said, commission rates should not be the sole reason to change brokers, especially for a newer trader. Until a trader has a command of the language of trading and a complete understanding when and how to use the different types of orders, a good broker can keep that trader from making mistakes when placing orders whose cost could far exceed any savings realized by a lower commission rate. If a lower commission rate makes more than a 10% difference in your bottom line as a trader, your trading methodology needs to be examined, not the commission rate.