I can't tell anything about your problem from your screenshot and JPEG because they are too low-resolution.
Zoom into a portion of the item of interest and trim your screenshots down to what you are actually showing; no one needs to see your whole monitor. Or post the .AI file. A screenshot is just that--a screenshot. No one can tell what's going on with all those fine lines, because of the moire pattern resulting from the pixel grid of the low-res screenshot image. So I'll assume that is the problem you are trying to convey: the fine pattern of curved lines looks different when you rasterize than it does when you view it zoomed in in Illustrator.
If that's the issue, it's normal. The same problems inherent in your attempt to display the problem here also apply to exporting as a raster image; the raster export will have to be of sufficient resolution to render all those fine lines using square pixels.
If you go nuts with this kind of design, you are going to encounter similar results in print if the fineness of all those hairlines is too detailed for the resolutions of the printing workflow. The image will have to be printed at a size large enough to be rendered by the size of the halftone screen that is to be used in print. The imaging device doesn't draw curved lines that change tone. It turns on and off printer spots to build halftone dots of varying sizes.
Any time you impose a grid over a design like this, you create a moire pattern due to the interference between the resolution of the line work and that of the grid(s).
So on screen, at any zoom, you have a fixed-size grid of square monitor pixels trying to render a bunch of tiny hairlines. Zoom in and out and the apparent pattern of lines changes because at smaller zooms there are not enough pixels to render the hairlines.
In print, you are imposing at least two grids: The fixed-size grid of printer spots have to be used to construct the variable-size grid of halftone dots. If the halftone dots are not tiny enough to render the hairlines at the scale you want to print them, similarly disappointing results will occur in print. And at any given scale, the printed results will differ between printing devices of different resolutions.
TSince you are exporting to raster image formats try this select your art then go toObject>Rasterize then export the file.
You will know right away if you were successful using this approach.
However it would be nice to see an expanded screen shot of your apearance panel and layers panel
What effect do you have applied?
The vector elements in the exported file (.jpeg), look different as in the original illustrator file.
...and quite naturally so. Aliasing and interference are strictly phenomena limited to preview vs. your monitor vs. human perception, but not the actual file. If you realyl want to retain the patterns, you must properly construct them. No way around it.