They have a simlar story: Clouds of darkness threaten the Crystal Empire, Twilight must find some solution involving the Crystal Heart, and in the end, another character saves the day, and the Crystal Heart's power is unleashed, destroying the threat.
So, let's take a look at the differences:
In The Crystal Empire, the threat was caused by King Sombra, a dark, menacing, and extremely cruel and sadistic unicorn who enslaves ponies, takes great pleasure in killing anyone in his way, and would have tried to take over Equestria leading to basically WWII. Leading to the first of two confirmed deaths in FiM, other being a minor non-speaking villain (not counting fish, Pinkie clones and Plunderseeds). Sombra had the least screentime and lines of any villains - but was a speaking villain nonetheless.
In The Crystalling, the threat was a completely inanimate snowstorm caused by the destruction of the Crystal Heart, which was destroyed by Flurry Heart's out-of-control magic. As she didn't know what she was doing, she is by no means a villain, so the snowstorm is the closest thing to a villain this episode. Making this the first episode with no villain involvement (even Princess Twilight Sparkle could have counted the Plunderseeds, which in any event were created by Discord even though he had already reformed by now). Celestia, Luna, and Rainbow Dash try to fight it but almost get frozen (redeeming value to me being Celestia doing something for the first time since her fight with Chrysalis).
In The Crystal Empire, the Mane 6 set up the Crystal Faire to restore the Crystal Ponies' memories, where Twilight learns she must find the Crystal Heart. She searches King Sombra's castle, overcoming various traps set by the extremely intelligent unicorn king with a combination of her magic and the assistance of Spike. In the end, Spike gets the Crystal Heart to Princess Cadance, and she and the Crystal Ponies use the Crystal Heart to destroy Sombra.
In The Crystalling, Twilight tries to find a way to fix the Crystal Heart so it can continue to protect against the storm. In the end, Sunburst figures out that the Crystalling combined with a few spells - provided by Celestia, Luna, Twilight, and Starlight - can fix the Crystal Heart and suppress Flurry Heart's out-of-control magic.
The Other Missions
In The Crystal Empire, Celestia gives Twilight a "test", telling her only she can do what needs to be done to protect the Crystal Empire. Leading Twilight to think that if she gets help from anyone else in the direct solution, she fails the test. When she is trapped by King Sombra, she tells Spike to take the heart to Cadance, unwilling to sacrifice her friends, brother, and sister in law, as well as doom the crystal ponies to slavery. This turns out to be the real test, so Twilight passes, and is closer to becoming a princess.
In The Crystalling, Twilight gives Starlight her first friendship lesson to reunite with Sunburst. At first the reunion doesn't go as well as they thought, Starlight not wanting to reveal to Sunburst what she did, while Sunburst is afraid Starlight will be disappointed if she finds out he's not an important wizard. In the end, they reuinte and remain friends after saving the Crystal Empire.
Starlight Glimmer and Flurry Heart
Of course Starlight was absent from The Crystal Empire, and is the main character of The Crystalling. Many opposed her redemption and possible becoming a Mane 6 (which, it seems, will not come to pass because outside of episodes where the focus is on her, she has never been seen with the others), but many also liked her in this episode. Here she tries to reunite with Sunburst, all the while making extremely adorable faces. In my opinion, she saved this episode from being a more dull repeat of The Crystal Empire, and she is what opens up new possibilities for Season 6.
Then you get Shining Armor and Princess Cadance's baby, Flurry Heart, the newest character to the series. Aside from inadvertently causing the snowstorm, she could remind you of Jack-Jack Attack, and is considered to be cute. Now her magic is under control, which is good because it means something like that won't happen again.