Q:

What parts of D&D 5e's design heritage come from D&D 4e?

What are the significant elements that D&D 5e's design has kept from the design of D&D 4e (and which were unique to the 4th edition of D&D)? What parts of D&D 5e's heritage are from D&D 4e specifically, not part of their shared heritage from earlier editions?
This isn't a "what are the differences" question, like other questions we have here. This is a smaller question: What is the same?

A:

This question is fairly hard to answer because a lot of the things "unique to 4e" were actually playtested at the end of 3.5e, and so aren't truly unique.
But if we look at the defining characteristics of 4e, then I can name a few things that made the trip to 5e. Others might find more, though.
Short Rests and encounter powers
The option to take a few minutes to catch your breath and reset some of your powers, as well as heal some HP as long as you have reserve remaining, was one of the key features of 4e and it's still around in 5e. 4e used Healing Surges and 5e uses Hit Dice for the health recovery, but other than that it's the same trick.
Ritual Magic
An often overlooked feature of 4e, and I'm not sure if it's used more in 5e, but the option of casting magic with a really long casting time but without using any of your combat resources was new in 4e and personally I found it a really cool addition.
Overnight healing
Rather than HP-damage healing over a course of days or even weeks if characters are left to their own devices, players restore their full health every night. This reinforces the idea that standard HP-damage represents bruising, fatigue, morale and luck rather than actual wounds taken.
Some other parts where the idea was brought over, although the mechanics seem different:
Scrapping skill ranks
Instead of having ranks in skills (which caused a huge difference in numbers between a trained or untrained person) characters in 4th edition were either trained in a skill or not. This meant that the numbers were closer together. For investing more resources in a skill, instead of simply having a higher +X on your skill check, you would instead get other cool perks for it. Like rerolling, or entirely new powers.
Unified attack bonusses
4th edition scrapped the differing Base Attack Bonusses between various classes, instead giving all characters the same basic bonus to hit (which was then modified by attribute). While a Wizard's punch was still a sorry affair (due to lower Strength), at least he didn't have to roll a 20 to hit most of the time.

