The Bug:

In Bedrock edition, Allays have special, higher fidelity animations for their Idle, Movement, and Dancing that Java simply does not have. In Java edition these animations are more simple and less detailed, which is an odd inconsistency to have as Java edition still has all the high quality animations for other new mobs such as Wardens, Sniffers, and Camels.

This report will showcase the difference in these animations by showing a render of the Bedrock's version first, on the left, then an in-game showcase of Java's on the right.

NOTE - The speed of these GIFs may be off from their intended motion.

Animation 1: Idle

This one is more subtle; in Bedrock on the left, the abdomen sways a bit and the wings flap in a more realistic, angled fashion. 

Animation 2: Movement

The movement animation is more noticeable; the wings again flap in a more detailed way, and the whole body and head move and sway instead of remaining static like Java on the right. 
NOTE - Bedrock's movement animation has it's own parity issue; it lacks the increased wing fluttering speed from Java when Allays move very quickly that came of the original fix of this report. (MCPE-174606) Both the Java and Bedrock are missing parts the other has, here!

Animation 3: Dance

The Allay's Bedrock dancing animation VS its Java dancing animation is perhaps the most noticeable discrepancy. The subtle little wind up, and posed, flared out wings during the spin in Bedrock's is a far cry from the very simplistic model rotation Java's animation has. 

Notice about the history of this bug:

This bug was originally fixed in 22w42a. It fixed a lot of odd posing and clipping issues Java Allays experienced that Bedrock ones did not, and even added a movement speed factor for Java Allays that ended up surpassing Bedrock's (again, MCPE-174606). However, it had to be reopened because a notable animations were still not up to parity.