Yes, there is. And anyone who disagree should try and make an omelet out of a chick. The line is so obvious, that denying it is nothing but intellectual dishonesty. Though the line is fine, it is also undeniably existent in the same way the line between fried chicken and fried eggs is undeniable.
I would like to start my argument by stating that I think that a fetus is a living thing, so don't get me wrong there. With that having been said, I think that there is indeed a fine line between a fetus and a child. One can survive alone, the other cannot.
If you put a newborn and a fetus into the woods alone, neither would survive. They both rely on others to take care of them. The person who provides this care is primarily the mother, but you see in many species adoptive parents taking on the roles as well. Just because the baby is living inside the mother does not make it any less of a baby, it does not even mean it is more dependent, many babies come weeks or even months early and require no more care than a full term baby would.
Show me a child that can survive on their own! You can't, it there was... Then there would be no need to take of the child once it is born. We would let the child fend for itself. A fetus is as dependent on the mother as a child is. The fetus has a heart beat as much as a child.
This is a topic that is very sensitive for most people but the simple fact of the matter is that there is no difference between a child and a fetus. Once the fetus is formed it has roughly the same attributes of a child, except it can't live on it's own and is basically a parasitic organism.