Sefirat haomer is a time where we count the days leading towards Matan Torah and Shavuot.  If the purpose would be to simply account for the days then why is it we count the weeks as well?

Additionally, what in fact is the purpose of the counting? Seemingly there is a value to every single day but what is that?

The Gemara in Yevamot famously explains that Rabbi Akiva had 24,000 students who died during this time period.  The reason for their tragic deaths, the Gemara explains is because they neglected to show the proper respect to one another.  If that is true we are left with two additional questions: 1. How is it possible that such talmidei chachamim didn't show the appropriate respect to one another?  2. Why did their deaths happen specifically during this time period?

The Shem Mishmuel relates the following idea that helps answer all of these questions.  When showing respect to another, one must find a uniqueness that allows them to stand separate and apart from you.  What happens however, when you are so close to another you don't even sense that you apart at all? The Shem Mishmuel explains that it would be ridiculous for one part of the body to respect another part. They work together and without their contributions the work of the whole would never be accomplished.  The 24,000 students of Rabbi Akiva were so close to each other, they didn't see each other as a separate unit but rather another piece of the same "body". Therefore, no difference could be highlighted to enable and create that feeling of a need to respect the other.

So what was their mistake? And why were they punished now?

Building up to Matan Torah, one has to understand that as close as we may be as we build our community and become k'ish echod b'lev echod as the Jewish people were in the desert when they received the Torah, we must not forget that the Torah was not only accepted by a people but also by every individual.  As much as we are one, we cannot forget that we are individuals.  Maybe that is why we don't only count up and focus on every one of the days but also the weeks to show we are not only focusing on the individuals but also the collective as well.  As close as the students of Rabbi Akiva became, their mission in life was to always remember they are not only a collective group but a group of individuals as well.