Very simply put, 5e has about the same core as 3.5e, but is considerably simpler.
No skill points, so INT can be a dump stat again. Instead, you have skill proficiencies, which generally can only be selected during character's creation, or by taking a Feat.
No free Feats every 3 levels. Most of the classes, if they want a Feat, have to sacrifice an ASI (ability score improvement) to get it.
ASI comes every 4 levels, just like in 3e, but gives you 2 points that you can put into your primary attributes instead of 1. No stat can be improved above 20 this way.
Many spells (esp. those of Cleric, Druid, etc.) have Concentration that can be easily disrupted. You can only concentrate on a single spell at any given time, so boosting your entire party with Bull's Strength or Cat's Grace is no longer an option.
That's the most noticeable differences, off the top of my head.
