Speech, Science and Lockpicking - Science and Lockpicking are the most cited examples, so we'll cover this group first. The skills themselves did nothing. Instead, in certain situations the game checked your skill value against a break point. If your skill met or exceeded the break point, you were allowed to do something. If not, you weren't allowed to do the thing. The only value that mattered at any given time was the break point value. In other words, only one of the Skill points out of the 100 was important, and only in situations where the game was actually looking at the skill. The rest of the time, not even that mattered.
Guns, Explosives, Melee Weapons, and Energy Weapons - each point in these skills gave +0.5% damage to their weapon type. Over the course of leveling the skill to 100, this was a significant boost, but on a per-level basis, the damage boost was negligible.
Medicine, Repair, Barter, and Survival - much like the previous skills, the boost per level was insignificant. Unlike them, quantities made up for quality. The items these skills covered were common enough that more items could, and often did, accomplish the same thing without harming your character or supplies.
Unarmed - You thought Lockpicking and Science were bad? Unarmed did nothing. The damage of an Unarmed weapon did not change with your skill level, so there was pretty much no reason to invest in it beyond arbitrary break points.
Sneak - I have yet to see anything quantifiable on Sneak. Nobody knows exactly how this works. There are a lot of vague tips, but no hard rules with quantifiable things like viewing distance of targets, angle of their view, cover, etc.
The basic point of this is that the skill system used in New Vegas was a placebo. We made numbers go up, and it made us happy. I could throw out buzzwords like 'Sense of Progression', but that is literally just a complicated way of saying that it made us happy. And people are complaining that the system is gone.
With the new system, every point has a significant effect. A massive damage boost, the ability to open or do things we couldn't, etc. To me, asking for the skill system back is like asking to be treated like a child: praised for every minor accomplishment, even when that accomplishment amounts to nothing.
With the new system, every point we spend has a noticeable impact on the game. We're accomplishing the exact same things we were before, but without the Pavlovian system where we spend a lot of points for not a lot of gain. To me, that is nothing but good, but so many people hate it, and it boggles my mind.
So there you have it: my interpretation of the old system, and why the new system is so much better.