This question is very commonly asked, mainly due to the fact that a Ball and joint arm usually costs 3 times the price of a flex arm.
Ball and joints are made of Aluminum, they are stronger and will not budget even when extended to 24″ long. This allows you to shoot wide angle shots while reducing backscatter and still not worrying about current or movements changing the position of your arms. The downside is that positioning B&J arms requires opening a clamp, moving the arm and closing a clamp. This process usually requires 2 hands and while it is very simple, it still can be tedious.
Flex arms are very easy to maneuver, they can be pulled to any direction by simply moving them with one hand. They are plastic, lightweight, inexpensive and allow flexible positioning of your lights. The downside is the longer the arms, the more chance of them moving from current or simply the lights weight. This is why normally arms are used in a maximum length of 12-14″. Being shared like an arc when extended, this offers a 10″ distance between the housing and light. Much shorter than what B&J would allow. The locline arms in our new packages are strong, include an internal wire for securing the segments and are made of high quality durable plastic. This makes them a good alternative for photographers on a budget.