You're right, the weight of the bag remains the same, so there is no trick of physics here. The amount of exertion required to carry a particular weight is generally less when you can let your skeleton and legs do more of the work. When an item is directly over your center of gravity (e.g., a bundle carried on your head or in a bag that sits high on your shoulders), the weight is transferred directly onto fairly rigid parts of your skeleton (spine, hips, legs). Your leg and back muscles will have to do some work to keep your skeleton aligned, but not much compared to the work your arms have to do to hold the same weight out away from your body. When carrying an item with your arms, your skeleton is still supporting the extra weight, but considerable extra exertion is required from muscles in your arms and shoulders to transfer upward force out to the item.

The efficiency you gain by letting your skeleton rather than your muscles bear the weight is simple to demonstrate. Try holding a brick or some other heavy item in your hand with your arm fully extended away from your body. After a few minutes you will notice strain in your arm muscles. Then try moving your arm so that it is raised vertically, straight up from your shoulder. The weight is now being supported more directly by the bones in your arm, and you should find that far less muscular exertion is required to hold the brick in that position.

The reason why it feels different despite having the same weight is due to the fact that torque has an effect on that amount of applied force.

The best way to think about it is that it is not a simple linear support problem.A rotation problem is involves as well where you try to keep the object you are carrying from rotating your arm downwards. And the further out it is, the harder it is to keep it up.
A good way to think about it is in terms of a lever. If you give a weight a longer lever arm, it can lift something heavier due to the rotation effect of torque. Thats basically why depending on how you carry an object it will feel different, since it requires more applied force to prevent rotation. So carrying objects closer to you is easier. For instance, you might be able to lift a 70 lb weight close to your body, but if you try to carry it at arms length you could find it much more difficult.

I once had a question similar to this one which was answered by my teacher in the form of a question: "Would you rather have an elephant step on your hand while wearing a running shoe or a high heeled shoe?" The answer is mainly due to the concepts of pressure and surface area. If the elephant was wearing a running shoe it's weight would be divided throughout the surface area of it's shoe which would spread among your hand. If the elephant was wearing a high-heeled shoe most of it's weight would be directed down the heel on a single point of your hand --this would in turn hurt more.So to address the question of the bag. If you are holding the bag by a thin strap it may feel heavier than a wider strap on your shoulder. But maybe the straps are the same. So, this question also contains another element to it in which the rest of your body helps out. By holding the bag in your had at your sides the weight of the bag is still pulling down and you are only able to use the strength in your arm and hand to hold it. If the bag is hanging from your shoulder, it hangs down, and is primarily held up by your shoulder but also your back, torso, legs, etc. can help in alleviating the pressure of the heavy bag weighing down. It's amazing how the elements of the human body all work together for things we don't often think about.

Mostly this is due to how the weight is distributed. If you distribute the weight over a large area, then there is no strong pressure at any particular point. For example, a strap over your shoulder covers a lot mroe area than a simple handle in your hand, so the weight is carried easier.

Another factor to consider is how much padding (muscle or fat) is under the skin holding the strap or handle. Bare skin on bone is easy to pinch or bruise, which is why it's easy to get a black eye or bruised hand. But skin over areas of thick muscle or fat has a good cushion, which again helps distribute the weight over a larger area, so it's less painful.

The last factor to consider is your posture. It's hard to carry weight just from one arm because it always pulls you to one side. Carrying the weight in a backpack or on your shoulders helps keep the weight centered over your spine and hips, so you don't have to use muscles to keep yourself balanced.

Torque: the amount of force required to exert a rotational acceleration on an object is dependent upon both the magnitude of the force applied multiplied by the distance from the center of mass of the object the force is being applied to. So, in the case of the bag in your hand, gravity is pulling you over, so you have to exert more force to maintain your balance, hence more effort. On the other hand, if you've got the weight in a backpack, then it's quite close to your center of mass, and you can just walk around with it.

It is not only the weight or FORCE that makes a difference, it is the FORCE PER UNIT AREA that is important.
So if you have the weight of a car and it is resting on a 1 square inch pedestal, then the Force per unit area called the PRESSURE will be much higher on that 1 inch square area than if the pedestal is say 1 foot square. Basically the weight (force) is spread over a large area in the latter compared to the former