As of my understanding Partition By is almost identical to Group By, but with the following differences:
That group by actually groups the result set returning one row per group, which results therefore in SQL Server only allowing in the SELECT list aggregate functions or columns that are part of the group by clause (in which case SQL Server can guarantee that there are unique results for each group).
Consider for example MySQL which allows to have in the SELECT list columns that are not defined in the Group By clause, in which case one row is still being returned per group, however if the column doesn't have unique results then there is no guarantee what will be the output!
But with Partition By, although the results of the function are identical to the results of an aggregate function with Group By, still you are getting the normal result set, which means that one is getting one row per underlying row, and not one row per group, and because of this one can have columns that are not unique per group in the SELECT list.
So as a summary, Group By would be best when needs an output of one row per group, and Partition By would be best when one needs all the rows but still wants the aggregate function based on a group.
Of course there might also be performance issues, see http://social.msdn.microsoft.com/Forums/ms-MY/transactsql/thread/0b20c2b5-1607-40bc-b7a7-0c60a2a55fba.