Do search engines care if I use relative paths in my internal links instead of absolute paths?

<a href="/">My website root</a>
<a href="http://mysite.com/">My website root</a>

In my case it doesn't affect as far as traffic is concern but in gathering links, that is another story.

2-3 years ago yes it would have mattered. A big push back then was basically keyword stuffing which would include repeating keywords in meta tags, header tags, content, image tags, links, etc.

Back then using www.mysite.com (assuming mysite was a keyword you were trying to rank for) would have been big brownie points because it would have stacked with www.mysite.com/synonom-to-your-site-keyword increasing the chance of being ranked for that keyword. However nowadays the best practice revolves around a more natural way of building keywords so using relative links does not hurt your website rankings.

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