Contact Scores and Touch Points are designed to show you how active or hot the contact is.
Touchpoints are more of a long term measure of how many interactive sessions someone has had with your company. For instance, if they open and email, click a link and then look around your website, that would all just be one interactive session, even though they had taken multiple actions. There must be a 10-minute lack of activity before the next interaction would count as another touchpoint/session.
A contact score is different to touchpoints as it gives contacts a score for each and every individual action they have taken and can highlight active contacts to you in the short term. Some actions give a higher score than others, for instance just opening an email is 1 point where as completing a form online would be 7 points.
Using these two measures, touchpoints and scores you can see who is most engaged with your company in both the short term and the long term.