Are the two terms referring to the same person?
The traditional role of a stockbroker is to arrange for the buying and selling of stock by finding buyers and sellers at an agreed upon price. The broker does not purchase the stock for himself but merely arranges for the stock to be traded.
A trader is one who purchases stock with the hope of selling it for a gain. The trader will use a broker to help with the purchase and sale of a stock.
Traders trade for a living, stockbrokers tell people to get involved in trades for a living. To be employed as a trader, you need a proven track record of being able to consistently make money. To be employed as a stockbroker, you need to get licensed but you don't need to prove you can consistently make money.