The choice of rendering engine will certainly affect the appearance of HTML when converted to PDF, so it’s a point worth considering.
Webkit is fine, as used in Chrome and Safari. But you probably wouldn’t want to select this if your user base was heavily dependent on another browser.
This is sometimes the case in corporate environments, where users choice of browser can be limited. South Korea is another example, where the vast majority of Internet users are obliged to use Internet Explorer (90%?).
FireFox uses the Gecko rendering engine, Internet Explorer uses Trident.
Pick a browser and PDF library that use the same engine, you’ll stand a better chance of producing a PDF that matches what you’d expect to see in your browser.
All the main web browsers use rendering engines that support HTML and CSS. Yes, even Micorsoft’s browser! 
Web pages can also include technologies that require browser plug-ins to support, such as Flash. You may want to ask if the PDF library support this too, and JavaScript, etc.