Main aspect is tire flexibility. This is given by tire design, rubber formula, casing tpi as someone else stated, and the amount of rubber used to construct the tire.
The more rubber the stiffer the tire.
The higher the tpi the more flexible the tire. When a bump hits the tire, the casing is responsible for holding the rubber together and avoiding the tire to explode like a balloon. The casing spreads the load among the threads near the hit spot. Some of the load fades from one thread to the next until some thread is rather unaffected. With high tpi casing, there are a lot of threads available, but they are also thinner, so they can stretch a little bit more and with more independence from the adjacent threads. These threads effectively absorb the shock. With low tpi casing it works the opposite way: threads are fewer and thicker, they stretch less, so they are less capable of absorbing the shock and rather transmit it to the rest of the tire structure. (Something like comparing thick denim against soft cotton fabric for t-shirts)
Regarding rubber, there are many formulas, each maker claims theirs is the best, but there is always a compromise. Soft rubber has better traction but adds rolling resistance and has less durability.
As per design, there are a lot of factors (casing tpi is rather a design decision), among others: ply count, thread type, etc. I have more experience with MTB (I only have one road bike and is the less ridden of my bikes), but the same aspects are comparable to some extent. In MTB the thread design affect how stiff the rider perceives the tire: if the spacing between blocks is as much as the width of these blocks or bigger, the blocks have more "range" to flex, cushioning a little bit more. This effect is also increased as the block height increased. A tire with smaller inter-block space and/or shallow thread gives the blocks less flexing options so the tire feels stiffer and with less rolling resistance.
The anecdote for context: I do not have much experience with road, but something similar happened to me with a hard tail mountain bike. I had installed some generic 26x2.1 tires, which where made of a thick and hard rubber. They here heavy but had very good traction both on pavement and offroad. These where HARD tires, they don't even lose their shape when completely deflated, and can stand the weight of the bike w/o air pressure, i.e. you wouldn't notice the tire is punctured if you just see the bike parked around...
The new tires were almost the same size visually, bot with lower and smaller thread. The rubber is way softer and the walls where a lot thinner. The tires are also very light, and foldable. (WTB Motoraptors).
As you stated, the bike feels a lot more comfortable, let's say it is more "forgiving" to your butt and spine regarding bumps and holes in the road. I felt the difference in my wrists also, the new tires gave me less vibration to the handlebar.
Both tire sets where used at about 40 psi, neither set had front or rear specific tires. The bike is a hardtail with some 110 mm air fork that is rather stiff and I use the bike to commute in a very hilly city with very bad pavement and also use the bike in trail mtb. I have even ridden some Downhill trails with this bike. The saddle is narrow and with thin padding.
