In the Harry Potter universe, the people in photographs and paintings can move and talk. You actually see characters interacting with the "people" in the pictures, particularly in the portraits that hang all over Hogwarts School. Then why can't Harry's parents photos interact with him ?
You identified the main difference in your Question. One is a Photo and the other is a Painting. Now we can only imagine the magic which goes into creating a Painting we do have this from JKR though.
Q: All the paintings we have seen at Hogwarts are of dead people. They seem to be living through their portraits. How is this so? If there was a painting of Harry’s parents, would he be able to obtain advice from them?
JKR: That is a very good question. They are all of dead people; they are not as fully realised as ghosts, as you have probably noticed. The place where you see them really talk is in Dumbledore’s office, primarily; the idea is that the previous headmasters and headmistresses leave behind a faint imprint of themselves. They leave their aura, almost, in the office and they can give some counsel to the present occupant, but it is not like being a ghost. They repeat catchphrases, almost. The portrait of Sirius’ mother is not a very 3D personality; she is not very fully realised. She repeats catchphrases that she had when she was alive. If Harry had a portrait of his parents it would not help him a great deal. If he could meet them as ghosts, that would be a much more meaningful interaction, but as Nick explained at the end of Phoenix—I am straying into dangerous territory, but I think you probably know what he explained—there are some people who would not come back as ghosts because they are unafraid, or less afraid, of death.
When it comes to photos it is something much closer to home to be able to have a moving image. Think of moving photos as videos without sound, they are basically the GIFs of HP, the difference being they are in newpapers and frames.
So a painting is really just memories from the subject and it's responses vary on how 'realised' it is. This would be much more complex magic than taking a photo (although I would suggest the cameras take videos and the magic with photos comes in putting them onto paper) with paintings it seems that the creation of them will allow them to talk. So you could also have a painting who can't talk, or like JKR said Sirius' mother who screamed out at most 3-4 different catchphrases I think?
To surmise Photo != Painting or Painting > Photo.
I always just assumed that there were different (magical) processes for making different kinds of pictures. Just like in the real world the difference between a black and white photo and a lenticular 3D color picture. We se at least 3 different kinds. The plain pictures with no associated magic, the animated pictures that are just animated with no interaction and the fully interactive ones that can talk and interact.
Portraits in Harry Potter are made sentient using enchantments placed on the paintings by the painter. Most of the talking paintings actually know little if any of the life of their subject. The exceptions are mostly the Headmasters' portraits that are painted early and are kept so the Headmaster can add subtleties and intricacies they deem important for their portrait.