What is the difference between "encumbered with" and "encumbered by"?

with implies a load that you have to carry (the load itself is the encumberence). by means there is an impediment that is blocking your movements (the impediment is not inherent to you, it is not of you and what you are doing).

One can be impeded by uneven ground and randomly tossed objects on the ground. You cannot be impeded with those things; they are not part of you; they are external to the action you are trying to perform.

No real difference. You may find "with" is used more with real objects and "by" with people or ideas.

I was encumbered with a heavy backpack.

Hillary is encumbered by Bill's womanizing.

There is no difference at all. You can use whichever expression you like.

I'm not sure there is a difference, the use just depends on how the sentence flows. Someone could be encumbered 'with' or 'by' a heavy bag or shoes or whatever.