A pro will likely come by with a good answer. But I would think that if those beams have stood for 100+ years without sealing and exposed in an unconditioned barn, they likely do not need sealing to be in your conditioned home.

The beams that are sealed have less splinters and you can kind of dust them. The beams that are not sealed are very dusty and dull. you cannot dust them because anything that you run across it will snag on splinters, including your hands. I’m not concerned, obviously about the strength of the wood, but more or less keeping the integrity of the wood by putting a sealant over it so that it can be wiped down and not dull and dusty. Does that make sense?

The first photo is an unsealed beam, the second photo is a sealed beam. You can see the first photo how the quality of the wood is versus the second photo.

Personally, I prefer the look of the unsealed beams. However, I think the combination of the two looks a bit incongruent. So, perhaps sealing would address that.

It might be that the vertical areas are what was left exposed, and developed a patina. The horozontal areas would look different because they were covered with flooring. Those areas might have been left a bit more unfinished to start with, the builders knowing the ugly face can be hidden by the floor. Some of it might be those surfaces dried out more by comparison to the exposed surfaces gaining a patina.

Depending on where you are and who built the barn, it's possible that after the barn was built and before it started being used, they used linseed oil on the exposed beams and sometimes the floors. That could contribute to the patina that developed on the exposed beams.

Some vertical shine can come from if there was a lift there, with things like bales rubbing against it. You also have wear areas on at least one of your vertical beams that shows lighter than the rest of the beam. That's likely from someone moving around it a lot.

But the rougher wood can be dusted by using a vacuum wand with a brush attachment.