Impulse is the change in momentum. It doesn't factor in velocity and mass of the object. Instead, it focuses on just one object. If you are going to use:
final velocity = initial velocity you must know the velocities for the objects at each stage.
Impulse allows you to determine the change in momentum. Whether momentum has been increased or decreased of one particular object will determine if the object increased or decreased in velocity. Knowing the impulse will allow you to find the velocity (assuming no mass is lost). As a result, to find the final or initial velocity, impulse will be most relevant/useful and would just require some simple substitution and algebra. For example, in the problem you stated, the wall doesn't have a specific mass, nor would it move from a collision from a ball. As a result, using impulse will allow you to find the change in momentum of the ball, hence allowing you to find the final or initial velocity.
Impulse = change in momentum = velocity final - velocity initial
As you can see there is only one object considered in impulse. This would mean you do not need to know anything about the other object. However, note that the change in momentum (impulse) always add up to zero in an elastic collision (where no energy is lost). So this can also be helpful if the question only gives you the impulse for one of the objects.
Finally, almost always use impulse if you are given a force or time quantity. If only the velocities are known, then use the equation you mentioned. It all depends on what quantities you are given and working around that to find the answer.