You told us you are using a .gif file in After Effects. You didn't tell us what kind of file you're rendering out of After Effects. It's important.
Sorry, I rendered it as a .mov. and also tried exporting is using Media Encoder but couldn't find .gif in the export options. I rendered it also with the default settings and also with Draft settings and lossless mode. I have several versions of the same thing. I also exported it as Photoshop layers at one point.
First, 16 fps is a very high frame rate for a gif. Second, the default uncompressed file from the default Output Module settings is going to be huge. Third, if you want to deliver a gif what you need to do is load your rendered uncompressed movie into Photoshop and then delete every single frame/layer in the movie that is not needed, adjust the timing of each individual frame and then export your animated gif from Photoshop. That's the only way to maintain quality.
For example, you are designing a ad banner that shows a product and a logo for 4 seconds and then the product zooms off screen and another product comes on along with a new tag line and holds for 4 seconds and then moves off and repeats. You set up layer 1 in photoshop with a duration of 4 seconds, then use the next 5 frames to do the transition giving each frame a duration of .1 seconds, then you hold the next layer/frame for 4 seconds, then use the next 5 frames to do the transition back to the first frame. This means there are only 12 frames in your animation, the first and sixth frame are 4 seconds, the other 10 are .1 seconds. That's how you get an efficient gif animation. You can't do it in AE directly.
If you are trying to do something else let us know.
Thank you very much for your reply, this was really great advice, I haven't compromised on quality and the animation still runs smoothly. I'm really pleased with how it turned out!