What is the difference between "left/right side" and "left/right-hand side"?

Right and left are variable directions. Right-handed and left-handed are fixed attributes.

Right and left are indications of orientation based on the viewer's current position.

The frame of that painting is cracked on the left . (But when viewed from the back, it is broken on the right.)
The tree on the left must come down. (But where are you standing when you give this command?)

I am left-handed. Whether I am viewed from the front or back, my left hand is always the same. (I think that right-handed people may have a similar experience.) Clothing and things that are based on human anatomy maintain their left-hand/right-hand orientation no matter how you rotate them.

Additionally, certain objects that are dependent on human control are given a handedness, such as cars and boats. The left front tire is the one on the driver's left hand side in front of his position when driving. Even when the driver walks around the car, the left[-hand] front tire is fixed in that spot, even though it may be on the left or right of the driver as he looks at it from various vantage points.

Similarly, the starboard side of the ship is to the captain's right when she is on the bridge, facing the prow. When she is on the prow, facing the bridge, starboard is still on the boat's right, even though it is on the captain's left.

Right-hand screw threads always turn in the same direction. When looking at the axis of the threaded object, turning it clockwise (top moves toward viewer's right hand) the threaded object moves away.

Finally, in the theater, sides are locked in. Stage left is to the actor's left side when he is facing the audience. If he turns his back to the audience, the stage's left-hand side does not shift, even though it is now on the actor's right. Move to your left is not necessarily the same as move stage left. Both the actor and the stage carry their handedness with them and instructions need to be clear which is intended.

(Since all the world's a stage, and actors come and go, I think stage directions will usually prevail.)