How did you set up the styles in InDesign?
What is the font used in AI / ID?
It doesn't appear that the font has been embedded properly in the ID file and is substituted.
I'm thinking that when you exported to PDF, you used a PDF Preset (located at the top of the dialog box) that did not embed the fonts. Try using High Quality Print or PDF-X/4.
My tried both, "Interactive" and "Printing" ( "Printing" i used High Quality Print or PDF-X/4. but it made it even worse some no background images create a white outline they didn't have when exporting with "Interactive").
The font I used it is "century gothic".
When i zoom into the PDF 190-200% the typography kicks in, but it displays with jagged edges.
I attached an image with the .ind and pdf differences and how tracing look how it should.....
The ease way it's tracing the text, but make my job a lot more annoying and the file probably will get a lot heavier.
I'm not sure if the problem can come from the styles, i don't think i did anything out of the box, if you could be more specific about where could be the problem i will try changing stuff.