Gif export has snow/moving white dots not wanted

Here are some basic rules for Animated GIFs. 

1. There should be no duplicate frames in the animation. If you need to rest for 4 seconds on a frame with no movement, set the duration of that frame inside Photoshop.
2. You are limited to 256 total colors. Be very careful with your color palette. 
3. The maximum frame rate for an animated GIF should be 15 fps. 12 fps is fast enough for most animations. 12 fps works for Buggs Bunney, so it should work fine for you.

The most efficient workflow for creating Animated GIFs in After Effects is to start with a 12 or 15 fps composition that is a few seconds long. Make sure the Project settings are 8-Bit. This is millions of colors, but it will show banding issues much better than a 16 or 32-bit project. Start adding the layers and begin the animation on the first frame. When the first move stops, and you want to pause for a while, as you would in a banner ad, and you have set your last keyframe for the move, move forward one frame and set another keyframe for every layer that is animated. Start your second move, and repeat. When you are done, you have a comp that contains movement, then one frame with no movement, then movement, and so on. 

If you want a perfect seamless loop, the sequence's first and last frames of the sequence must be identical. Then you go to the last frame of the animation (identical to the first frame if you want a seamless loop), move back one frame, set the out point of the work area by pressing the N key, then Trim the Composition to the Work area. If you do not need a perfect loop, set the work area to the last frame of the animation.

When you are done, add the comp to the Media Encoder and pick the Animated Gif template that matches your comp. Transparent BG and No transparency are the only real choices. Export your animated gif.

Here's the critical final step. Open the Animated Gif in Photoshop and make sure that the Motion workspace is active. Move Select the frames you want to extend and change the frame duration. Here's a screenshot of that from a previous post I made.

Here is the thread.

Then save using Photoshop's File/Export/Save for Web Legacy menu. Set the final color palette and loop options. 

Work that way, and you can have an animated gif that lasts for 2 minutes with four one-second transitions between scenes that is only 64 frames long.  

If that workflow does not solve your problem, export an image sequence from After Effects, load the image sequence into Photoshop and create your animated GIF there. You will have complete control over the color palette that way.

PS*** Changed export preset to Animated Gif (match source) in export settings and still the same problem