Pegacorns: ponies of originally one race given a horn and wings.

Alicorns: ponies born with both a horn and wings. 

Both Pegacorns and Alicorns are immortal*, meaning they can live for more than thousands of years. Their height and magical/flowing hair depend on how powerful** they are, not by how long they live. One of the only differences between Alicorns and Pegacorns is what race they are born as and how they are able to retire***. When Pegacorns retire, they lose the wings and/or horn they were given, and they're able to age again. Alicorns, on the other hoof, are unable to lose their wings or horn**** and may never be able to permanently retire because of it.

I headcanon that Celestia and Luna are Pegacorns. Celestia was born a unicorn, while Luna was born an earth pony. They were also unrelated, but in my full headcanon, Luna practically got adopted by Celestia's original family when they were mere fillies. Both Twilight and Cadence are also Pegacorns. So basically the entirety of Equestria was watched over by Pegacorns until Flurry Heart's birth. Finally, an Alicorn born after possibly millions of years. The poor baby doesn't know yet that she may be unable to retire like her mother and aunts. 

Bonus Headcanon... So what about Twilight's friends and family? Since earth ponies have the longest lives and unicorns seem to maintain their youthful and majestic looks, Applejack and Rarity would live the longest out of their friends. Rainbow Dash and Pinkie, being as reckless as they were in the past, may pass the earliest. As for Fluttershy, she becomes immortal herself. What, how? I feel like Discord wouldn't dare to imagine Fluttershy going away, so he grants her a new Draconequus body. After she hesitated at first, the creatures of Equestria began to nickname her "Mother Nature." For Twilight's family, they don't become immortal in any way and they soon pass as well, including Shining Armour, much to everyone's dismay. 

*Immortal: We (or at least I) don't know how long Alicorns and Pegacorns really live. It's possible that another difference between the two is that Alicorns live WAY longer than Pegacorns, but I'm not gonna jump too far into that. 

**Powerful: Power can mean how vast or strong their magic is or their love is. But since Cadence isn't too tall, I don't know for sure if love is power. 

***Retire: Retiring from the throne, whether to give it away or to live a normal life.

****Alicorns are unable to lose their wings or horn: Unless they have the help of Discord-esque magic. However, I don't know how long said magic lasts or if it effects the alicorn's lifespan.

Thanks for reading! Questions, suggestions, or other feedback is appreciated!