Her big draw is manipulating enemy locations. Her Q is a REALLY slow skillshot that is going to make you take 1.8 AP scaling damage if it stuns you (from Q + E). Her wall limits where the enemy can go. From a more advanced perspective, you can actually directly manipulate the enemy's pathing depending on the location and angle of the wall. Her R is a lot more obvious with its flashing and high sustained damage.
Using Q, W, and R in combination to control the enemy's movement is something that's not necessarily unique to Anivia, but is definitely something she's capable of doing better than most of the cast.
Example situation: You're in mid lane, and the enemy wants to get to safety under tower. Some options you have (depending on your location and your enemy's location) include:
-Throw your Q towards the small gap between tower and jungle, and put your R in the other one. Enemy has to get hit by at least one of them.
-Throw your Q towards the center of the lane. When the enemy runs to the gap between tower and jungle to avoid Q, you put your W up behind him. His pathing makes him run around the tower and get hit by your Q.
-Put your R down smack on top of the enemy. As he's about to get out of it, put the wall down behind him, forcing him to run all the way back out through it while slowed.
Her burst is nice, but it's not nearly as reliable as it was when you could do ER and have E do double damage just from the first tick of R during travel time. Even if you land both parts of Q into E, her burst (1.8 AP) isn't as high as other famous burst mages. For comparison:
-Veigar - 2.35-3.1 AP (depending on target health) from QWR from similar range as Anivia
-Lux's - 2.25 AP from QER (without any autos) that she can do from ~1000 range
-Annie - 2.3 AP (without any Tibbers ticks or autos)
-Vel'Koz's - 1.3 AP magic damage + 0.5 AP true damage on non-ult QWE (similar to Anivia) but ALSO has 1.75 AP true damage scaling on just his ult.