The first difference between these two is that the zaibatsu are a historical phenomenon while keiretsu still exist. The zaibatsu were dissolved in the years of the American occupation and the keiretsu sprang up by the 1960s. These are both conglomerates of various businesses that are controlled more or less centrally.
The major difference between the two in terms of their structure has to do with what ties them together. The zaibatsu were essentially family businesses. A zaibatsu would be made up of firms whose leaders were related to one another in some way. By contrast, the keiretsu come together for business reasons. They tend to be centered around a bank that sits at their core. They are tied together by the banks and by business interests, not by kinship.