A person can be leaning over or bending over to pick something up. Bending over gives the image of the body bending at the waist and going forward toward the floor. Leaning could be to the side or toward the front. A book or a ladder or inanimate object can be leaning over, (angled) but cannot be bending over. An idiom "bending over backwards" means doing one's utmost to achieve something: I am bending over backwards to try and say I am sorry for our argument.

If there is a difference, bending, you can be to search for something and bend refers to an object

Hi Peachey, thank you for your answer.

Hi Tim, there is a big difference if there's an object in the sentence. If you lean over a railing, then your upper body is forward, and probably supported by the railing. If you bend over a railing, there are two possibilities: the same as lean over (except your body is clearly folded forward), or you grab the railing and bend it out of shape.