There is no difference in meaning, but there is a difference in emphasis. If you want to emphasise that you are referring to your children and only your children, then you might say, "I am only going to be responsible for looking after my own children", meaning I won't be responsible for any other children. I am only going to be responsible for my children has the same meaning, but less strong emphasis.

To own something ie I bought a car so I own it. Children could be your own but not owned. Confusing. Ie. They are my own children but I do not own them.

The emphasis answer is correct-- the usage is most often like this: "You will need your own pencils for this test" Meaning, the testing center won't provide you a pencil, you will need to bring your own (see!), or you will have no pencil. It would be unclear if it said only "You will need your pencils for this test" Another example: "You will need your own sheets at this hostel" Meaning, the hostel has a mattress, but you will need to bring sheets that belong to you, because the hostel has none. Hope this helps!