They're used in different places. group by modifies the entire query, like:
group by
select customerId, count(*) as orderCount from Orders group by customerId
But partition by just works on a window function, like row_number:
partition by
row_number
select row_number() over (partition by customerId order by orderId) as OrderNumberForThisCustomer from Orders
A group by normally reduces the number of rows returned by rolling them up and calculating averages or sums for each row. partition by does not affect the number of rows returned, but it changes how a window function's result is calculated.