If the only thing you have is one instance of the competitor's name on one page, whether it's in a link (nofollowed or otherwise), it's really unlikely you'll show up anywhere useful in the SERPs unless your competitors are spectacularly bad at SEO and
have no business partners, never get mentioned in the press, never issue press releases, never get reviewed on anybody else's website, have no blog, don't participate in any social media, etc.
That said, yes, AFAIK you get "credit" for the words that appear on your pages, including words that are in link anchor text pointing at other pages. I've never seen anything anywhere to suggest that somehow you won't get credit for the words on the page just because they are anchor text for a link that's been nofollowed.
Be careful about linking to direct competitors, though. Nofollow only affects search engines, not human visitors. Do you really want to take a chance that your visitors will click through to your competitors? Once they get to the competitor's site, they may not come back to yours. Personally, I would never link to a direct competitor from a site where I'm trying to make a sale or generate a lead. But maybe that's just me.