Why will a golf ball accelerate faster than a bowling ball when struck by a golf club

I think you mean velocity. There is only a downward acceleration (gravitational acceleration) for bodies in free-fall, and their acceleration would be the same. So I'm going to assume velocity.

Using the idea of conservation of momentum, when you swing a a golf club, there is momentum of F*dt. where F is the force and dt is the time of impact. Once you hit the ball, momentum is transferred. Another equation for momentum is m*v which is mass times velocity.

Modeling this we see:

F*dt = m*v

So the larger the mass, the smaller the velocity. Smaller the mass, the larger the velocity. That's why a golf ball's initial velocity is faster.