What settings are you using now? Looks like something that is either converting to CMYK, or it's happening during flattening (the white lines are flattening artifacts called "stitching") because the blend space is set to CMYK instead of RGB, or both.
Now here's what's weird. I switched to Document RGB and it gives warning before exporting to PDF (of RGB/CMYK conflict). But the below is correct (except stitching still showing up)
So, despite the color conflict it came out right whereas before (with no color conflict) it exported incorrectly.
Will the stitching be printed or this is just a screen rendering issue?
OK, you have both problems.
Is this going to print? Your saturated red is probably out of gamut for most devices (certainly for the SWOP profile you've enabled) so won't be reproducible. Is your black spec'd as rich black or 100K? Were you told to provide a PDF/X-1a, or other flattened format, file? The stitching artifacts will disappear on a press, but it would be better to provide an unflattened file if they will accept it.
If they will take it, I would suggest you export using the PDF/X-4 setting, but you should still expect a loss of saturation in the output from the press.
And for good measure, though it won't affect the output of a color PDF, I suggest you make sure your Appearance of Black prefernce is set to Display All Blacks Accurately and Output All Blacks As Rich Black. The first prevents you from seeing 100K as darker on screen than it will print, and the second insures that 100K will print as a solid on a monochrome printer and will export as 100K in a grayscale PDF.