Force vectors
The idea of force vectors is to find out how much force is being or needs to be applied to a vector. For example if you have two people pulling a boat up a beach at different angles, you can find out how much force each person needs to apply for the boat to travel in a straight path, or vice versa (what angle is needed with given force).
Free body diagrams
In a free body diagram you are simplifying another picture. This visual representation makes it easier to follow what you are doing and the forces being applied to each object and the direction of those forces. This makes solving force vectors much easier, as you aren't constantly trying to annotate around a sketch. There is only a dot for the object and a line in the direction of each force and an arrow to show whether that force is creating compression or tension.
![Picture](/uploads/1/3/7/6/13768163/1404430.jpg?1354150402)
This free body diagram represents the boat pictured above.