Some dictionaries state that the two expressions have the same meaning, but there are some situations where you would use one but not the other. According to Cambridge Dictionary (and I am aware of this nuance in everyday life) to take someone's word for something means:
to believe that what someone is saying is true
When my son said he did not break the neighbour's window, I took his word for it.
There is often the nuance that someone believed someone else without checking; it does not always imply implicit belief.
Take someone's word for it
However to take someone at their word means (my emphasis):
to believe (someone) without question and act according to their words
When my boss said it was time to change the furniture in the office, I took her at her word and ordered new desks and chairs.
Take someone at his word