Having finished all four routes:

If you're only going to play one route, play Verdant Wind. If you're going to play two, play Verdant Wind and Crimson Flower. If you're going to play three, add in Azure Moon. Silver Snow is entirely redundant.

Azure Moon is the most "Fire Emblem" of the routes. It's got a deposed prince fighting an evil empire. About the only thing it's missing is a dragon as the final boss. It's also a fairly unsatisfying story -- Dimitri's character growth is handled poorly, and the actual villains of the story are just kind of forgotten about after part one.

Crimson Flower is a very flawed route. You can tell it's unfinished -- it lacks the animated cutscenes the other routes get, it's four chapters shorter than the other routes, and it just sort of ends with no resolution (other than an epilogue panel promising the real villains will be dealt with in the future). That being said, it's an interesting chance to play as the villains, even if it's not necessarily the best execution of that concept.

Verdant Wind is hands down the best route. You get actual backstory on the world and the major conflict, you get to actually finish both major conflicts in the same route, and the ending chapter isn't a stupid, out of nowhere "twist."

Silver Snow is Verdant Wind but missing a chapter, no real "lord" character, and a stupid final boss. Only play this if you're a completionist.

As a final note, Byleth's lack of voice lines is really awkward in some points, mostly the romance bits at the end. I sort of get why they made this choice, but it's a bizarre one, and it really hurts Byleth as a character and as a story vehicle.