The children are age 4 and 1

4 is the only number which can be a product of 2 different combinations of 2 numbers below 6, while one of these combinations consists of 2 identical numbers.

We need 2 combinations because the information of the product of their ages wasnt enough information.

One of these combinations has to consist of 2 identical numbers because the information that there is one child that is older than the other one has to eliminate that combination.

The only product this can refer to is 4 with the combinations 2*2 and 1*4.