Pros of 5e skill system Easy to remember skill bonuses. If 1 skill is +4 for Dex, all skills for Dex are likely +4 if proficient. I don't have +8 Stealth and +4 Sleight of hand, etc. Simple level ups. I don't have to spend skill points. Thus, it doesn't take a 30 minutes to level up 1 character. It gets the job done. If done right, there is still a considerable enough difference to make a distinction between proficient and not proficient. Stabilizes difficulty levels so you keep the same basic difficulties even as you gain higher levels. With previous systems, of you were unskilled, a moderate difficulty was brutal. You get like +1 or +2 against a DC 15 while a skilled person might have +8. Try being a mage with +2 for Athletics who needs to make a moderate jump versus a fighter with +8 or +10. Mage would fail like so many times. When it should be 50/50 for unskilled, it was 50/50 for skilled.
Cons of 5e Not as diverse with skills. You don't have someone with really high Arcana because they focused more in that, excelling in one skill far above the others. They might have +8 in Arcana, +5 in History, +3 in Religion, +9 in Persuade. Other systems vary the skills more.
I'm out of time, but I'm sure you'll all give more cons. 😁