I see several problems.
First, your comp size??? you should be working with Standard HD comps at a minimum unless you know exactly what you are doing and can explain why you would want to use something other than a standard HD comp for your project. You should also pick a standard frame rate. If your composition settings show Custom you had better know exactly what you are doing and why you are not using standard video formats.
Second problem. Look at the thickness of the lines in your graphics:
Judging from the size of your screenshot you are working with a HR display so looking at things at 200% is probably OK. The lines on all of your icons are about one pixel wide. That is going to be problematic when working with video for several reasons. The first is that the thin lines must be perfectly lined up with the pixel grid to keep them from being reinterpreted. A single pixel wide line, even if it is a vector, when it is not perfectly lined up on the pixel grid will become two gray lines. Movement will make the edges flicker. Thin lines and motion graphics do not go well together. Here's what I mean. This is a rectangle shape layer with a single pixel stroke. It has been positioned so that the vertical lines align with the pixel grid but the horizontal lines do not. Imagine how this would look if it were moving with the line constantly changing size and color.
If you are going to be creating very thin lines in Illustrator then you must make sure that the lines are precisely lined up with the pixel grid by turning on snap to pixel and pixel preview in AI. The layers also need to be precisely lined up on the grid in Illustrator and the canvas and the shapes must be an even number of pixels wide and tall or you'll be dealing with fractional position or anchor point values in AE. The other problem you have is that you are drawing black lines against a white background with the RGB values of both all the way to the max. MPEG compression is going to hate that and throw in compression artifacts all around your icons, especially if they are moving. If you are going to use white as a background then you've got to pull thin black lines up to about 16 in 8bit for the compression to have a chance of making good pixels from them. If you want to use fully black lines against white background then you need to pull the white down to about 235. When you understand how compressed video treats pixels then you will figure out why this is so. Sorry, them's just the rules for working with video. There is no getting around them.
The last problem is the way you have rendered your video. You should have been using the YouTube HD preset in the Adobe Media Encoder. Instead you rendered to 480p so that is the maximum quality available. This means your project was scaled down, but then when it is played back full screen is scaled way up making everything soft.
You need to study up on rendering and make sure you understand exactly what you are doing and why. If in doubt always go HD 1080 X 1920 29.97 unless you live in a PAL country (Europe) with 50 cycle electricity instead of 60 cycle and 25 fps video standards.
thank you for getting back to me so fast.
Following your suggestions I decided to start from scratch.
Are the comp settings bellow what you are referring to when you say "If in doubt always go HD 1080 X 1920 29.97 unless you live in a PAL country (Europe) with 50 cycle electricity instead of 60 cycle and 25 fps video standards."?
Im in Australia.