Morgana's snare has a larger hitbox and a longer cast animation. It also needs levels before the snare lasts longer than Lux's. Morgana's snare is on a shorter CD than Lux's. Lux has better poke damage with her AOE, Morgana has better waveclear. Morgana has lower mana costs and better sustain through her spellvamp. Her auto attack is also very short ranged. Morgana's shield blocks magic damage and CC while Lux's shield is bigger, aoe, and can block all damages.
It really depends on what you want out of the champs. Morgana can roam better due to waveclear, better all in potential due to ult and spell vamp, and is easier to play/more reliable due to lack of skillshots. Lux is better at stay safe, is better at being a lane bully with poke, and can snipe objectives with her ult.