PDF does NOT support rich text tooltips (e.g. underline, bold, etc.) at this time. In other words, PDF does not support all the formatting you've done; it's not an iGrafx limitation, it's a PDF (Adobe Acrobat) limitation.
However, you can turn on 'Print Notes' in the File menu's Page Setup command, which will output the tooltip and a page of all the notes at the end of the diagram. The formatting will be included in the separately published Notes because iGrafx can control how the text is output when it's not in a PDF comment/tooltip.
Do the following to enable a separate page (or set of pages) in your PDF document that have the Notes published with the formatting included:
From the File menu, choose Page Setup.
Click on the Options tab.
Check the 'Notes' and 'Shape Text' options under the Print section.
Publish to PDF as usual (File > Publish As > PDF and use the wizard).
On a related note, please be aware that the lines break (have an "Enter" or "Return" to craete a new line) based on the width of the area they are being displayed within. You’ll notice that if you widen the Notes Window, the line break points will change. Then if you look at the tooltip in iGrafx, it will be different since the max width of a tooltip is fixed. The same is true in the .pdf export. The Notes printed at the end use the width of the page (if there is no hard line break), while the PDF tooltips use their own max width. In both cases, hard line breaks (an "Enter" that has been placed there explicitly by whomever typed the text) are recognized.