RewriteRule /linklist.html /links.html [R=301,L]
Of course, your code above redirects www to www and non-www to non-www. I would add protocol and hostname to the target so that both requests end up in the same place.
I would also escape the literal period in the rule pattern.
Make sure that all rules that are like that one are listed before your site-wide canonical non-www/www redirect.