What is the difference between centripetal acceleration and acceleration due to gravity?

As far as an observer is concerned, there is no difference.  If you were in a big space station that rotated to create the feeling of gravity, you could not tell you were not on earth without looking outside.  It must be big so there is no significant difference between the radius to your head and feet ===>  No significant difference in the centripetal force on your head and feet.

Weight is a force so it follows F=MA  Yes EXACTLY the same thing.

Normal Force is the force at right angles to a "surface"  Force of Gravity = Normal Force for a horizontal surface.

For a sloped surface  Normal Force = sin angle of slope * Force of gravity

The acceleration due to gravity really is just what it says, namely the acceleration that a mass will experience simply by being within the gravitational field of the Earth.

The force applied by gravity however is not quite the same; it is equal to the mass of the object multiplied by its acceleration. So that's the key difference; the force multiplies the acceleration by the mass of the object.

This property is laid out by Newton's Second Law, given by the formula F=ma where F is the force, m is the mass and a is the acceleration.

Force is, if you like, a description of how quickly the momentum of an object is changing and so it kind of makes sense that its all about pushes and pulls. For example when a tennis player hits the ball on serve you'lllll agree intuitively that the momentum of the ball is increased; the ball is 'pushed' forward.

You are completely correct in saying weight is equal to the force due to gravity; they are just two interchangeable phrases, both are equally valid.

You are also nearly correct in saying that the force due to gravity (the weight) and normal force are the same thing. The major difference however is that the normal force will be in the opposite direction to the weight. This is why an object, say a book, resting on a table doesn't move; its weight is opposed by the normal force applied by the table. These forces have the same size but opposite direction and so the net force is zero. Hence the book experiences no change in momentum and so remains as it was i.e. stationary.    

This is an example of Newtons Third Law; the famous statement that for every action there is an equal and opposite reaction. The action of the book, its weight, triggers an equal and opposite reaction from the table, the normal force.

Of course if the book were on a slope then the size of the weight would not equal that of the normal force as weight always acts downwards whilst normal force always acts at 90 degrees to the contact surface. Hence the normal force would have a size equal to the component of weight at 90 degrees to the surface; it would still oppose this weight component though.

Centripetal acceleration is the name given to the acceleration of a object towards its centre of rotation. So by definition only rotatory objects can posses centripetal acceleration.

The acceleration due to gravity can be centripetal. For example, orbital satelites rotate about the Earth and so their acceleration towards the centre of rotation (the Earth) is just equal to the acceleration due to gravity.

However, just because gravity can act as a centripetal acceleration, does not mean that this is the only way in which it can operate. For example non-rotating objects are simply accelerated downwards by gravity; the acceleration is not centripetal.

Hope this helps.