When using a HREF link, it is important to attribute the link title correctly. Not too long ago, search engine users admitted they did not use title attribute in the equation because it was not used enough. This has changed since then, but there is still no evidence that link title attribution influences rankings. To test this you can include any nonexistent word that does not exist in a search engine index as a link title, the link will be indexed quickly, but if not just wait a little bit then check if either the linking page or linked page was ranked for that word.
Do not duplicate a link HTML tag, it is redundant and will not matter. Title tags are meant to provide additional information, if it does not provide more information, then do not include it. If the anchor links are conspicuous as to where they will lead then do not add the link title to the link. Link titles reduce the utility because it is one more thing that the user will have to look at.
A HREF link does not point to a different URL, but rather an ID or name tag. A HREF HTML is essentially a link to no where. It is an anchor text, that is all a HREF link is.