Sun or Earth???

That is a good question. I guess when it comes to religion, or rendering serving to God, science will change. During the Ramayan, everyone, including humans and Gods, wanted to render Shri Ram some service and help. To help Prabhu Shri Ram, maybe Surya Dev stopped the Earth from moving around it for as long as the vanar-sena needed to rescue Ram and Lakshman. This would not be completely against science, because the Earth didn't stop moving forever. After Shri Ram woke up, Earth once again revolved around the Sun.

You have asked for scientific explanation, so let me give that.
According to Modern Physics, there is nothing called absolute rest and also, there is nothing called absolute motion. Rest and motion depend on frame of reference. In other words, they are relative.
For example, I can see a table in front of me, on which my computer is kept. I see that it is at rest. Is the table at rest or is it in motion? Both are correct depending on where you fix your frame of reference.
If you fix the frame of reference on the Earth, then it is at rest.
If you fix the frame of reference on the Sun, then it is moving.

If you fix the frame of reference on the Sun, then the Sun is at rest and the Earth is moving. If you fix the frame of reference on the Earth, then the Earth is at rest and the Sun is moving. Both views are correct. Do not be shocked by reading this. It is a fact that both views are correct.

There was a time, when people thought that the Earth was rest and the Sun was moving. If it is true that it is correct to fix the frame of reference on the Earth, if we so wish, then why do we say that they people were wrong? They were wrong not because they fixed the frame of reference on the Earth, but because of following reasons:-
1. They did not have the idea of the frame of reference. They believed that Earth was in absolute rest.
2. They believed that Earth is at rest and the Moon, the Sun and other heavenly bodies move round in perfect circles. But, in reality, if you fix the frame of reference on the Earth, then the Sun and the Moon move in ellipses. But these ellipses are very close to circles, so even if we assume the paths of Sun and Moon as circular, then also we will get very accurate results, so that is fine. But the problem comes when you study the motion of a planet, e.g., Jupiter.
If you fix the frame of reference on the Earth, then the path of Jupiter consists of an ellipse with lots of smaller ellipses on it - these smaller ellipses are called as epicycles.
3. For studying the motion of the bodies in solar system, it is far simpler to fix the frame of reference on the Sun than to fix on the Earth. By fixing on Earth, the Mathematics becomes too complicated and we do not gain anything by the extra complexity - so the extra complexity is unnecessary.

Let me know if you need any further clarification on this.