Translation:Grandfather and I will take care of the tree.
In the context of this phrase ("Grandpa and I will take care of the tree"): In the sense of "Grandpa and I will provide for the tree/make sure that a tree is there", both "kümmern" and "sorgen" could be used: Großvater und ich werden für den Baum sorgen – Großvater und ich werden uns um den Baum kümmern. In the sense of "Grandpa and I will care for the tree/make sure that it grows well", "sorgen für" is usually more intense than "sich kümmern um". Besides, both words have other meanings and connotations where they are different from each other.
Caring about something means having feelings about it. You are afraid about what happens to it, you are happy about what happens to it.
Caring for something (a tree) means you make sure it has water, you make sure the soil it grows in is good, that it has enough light; you encourage beneficial funguses and plants to grow around it.