The main difference I noticed when I began a math postdoc is that postdocs are expected to act much more independently than graduate students. Math postdocs are viewed as (young but) full members of the research community, not as apprentices. You will need to choose your own goals and set your own deadlines.
One important change to make quickly is to develop (additional) professional relations with established researchers from other institutions. Some of these relations may turn into research collaborations, but others may just be discussions at conferences that lead to other discussions or to invitations to speak at other conferences or departmental colloquia. Keep in mind that you will need to ask for letters of recommendation when you apply for jobs again, and that others in your field will be reviewing your grant applications when you submit them.
In terms of research, as a postdoc you should also expect to always have multiple research projects underway, whereas many grad students only work on their thesis. You should also aim for projects that have a high chance of giving positive results quickly. Don't spend too long on famous open problems or on results that would take a decade to complete. If you apply to research positions, you will be judged on your productivity in the time since you earned a PhD, and you only have a few years in your postdoc.
The "clock": in an important sense, your vita begins when you finish your PhD. If possible, decide what sort of employment you want to have after your postdoc is over. Find well established colleagues you respect who took the path you are interested in, ask them for advice, and try to emulate them. You want to leave your postdoc with a vita that will make you as competitive as possible for the type of job you plan to apply for. A generic vita is unlikely to be competitive in the current job market.