Simple physics says that the cue ball will leave the collision with the object ball along the tangent to the contact point. This is a fairly simple result that might be taught in a high school physics course. It is mostly the result of conservation of energy and conservation of momentum. At least four assumptions go into the result: there is no sideways force at the contact point, no energy is lost in the collision, the balls are the same weight and size, and both balls are at the same height at the time of the collision.
On closer examination, at least one of those assumptions is wrong on every shot. That means that the cue ball almost never leaves the collision exactly perpendicular to the line the object ball takes out of the collision. How inexactly? That depends.
Due to sideways friction during the collision, several things happen. The object ball is pulled left or right of the line through the centers of the balls at the time of the collision (throw). That can be up to 6 degrees. This also causes the object ball to spin and the cue ball to slow down or speed up (a little) compared to the speed it would have without the throw. The direction of the cue ball is not changed by this effect but its spin is changed in the same amount that the object ball gains spin.
Energy is in fact lost in the collision. This will cause the cue ball to go forward of the expected line so the angle between the two ball paths will be less than 90 degrees. The angle change depends on the fullness of the hit and the "elasticity" of the balls. This effect is more noticeable with ivory balls because ivory is much less elastic than phenolic. You probably won't notice this effect except on some special shots.
The cue ball is often not the same size or weight as the object balls, due to wear. A heavy cue ball will tend to smash through the collision and a light cue ball will tend to bounce back. The change in the angle again depends on the cut angle.
Finally, if the cue ball is off the table at the time of the collision, which often happens when you are jacked up and shooting at a close object ball firmly, the path of the cue ball will appear to be ahead of the normal tangent line. In fact, the cue ball may be traveling along the real tangent line, but that tangent is to a point that is no longer on the equator of the object ball. An extreme case of this is when you shoot a jacked-up draw shot and the cue ball jumps forward before it draws back. In that case, the apparent change in cue ball path is 90 degrees.