Is there any difference between the Dance of the Fire god breathing and the sun breathing?

Question

The Kimetsu no Yaiba wikia refers to the Dance of the Fire god breathing and to the sun breathing as if they were the same breathing several times, and if they were different breath styles several times. For example, in these quotes it refers to it as if they were different,

he realized his own Hinokami Kagura (dance of the fire god) had unncessary movements, different wrist angles, footwork and breathing rhythm. Removing them enabled him to awaken the legendary Sun Breathing

Tanjiro Tamado

Both the Hinokami Kagura (dance of the fire god) and Sun Breathing are Breathing Styles that mimics the sun and replicates it with the user's movements, techniques and abilities

Hinokami Kagura

However here, it refers to them as if they were the same,

The Hinokami Kagura (dance of the fire god) is later revealed by Shinjuro Rengoku to be a legendary Breathing Style known as Sun Breathing

Hinokami Kagura

Hinokami Kagura (ヒノカミ神楽かぐら Hinokami Kagura?, lit. Dance of the Fire God) is a Breathing Style only known and taught by the Kamado Family, later revealed to be the first Breathing Style, Sun Breathing

So, are they different or are they the same?

Is there any difference between the Dance of the Fire god breathing and the sun breathing?

Answer 1

To answer this question you actually need to watch the later part of the second season for the show. Be mindful of spoilers

To begin, at the beginning of the entertainment district arc Tanjiro goes the house of Shinjuro Rengoku, who is the father of Kyojiro Rengoku, the Flame Hashira, to show his condolenses.(Please excuse some of the spelling, I'm writing in English) When he arrives outside the house he sees Senjuro outside the property sweeping. They talk for a bit and then his father comes out. After a bit of exchanging of words, things heat up when Shinjuro Rengoku realizes who Tanjiro is. What the most important thing to point out though is when Shinjuro Rengoku notices Tanjiro's Hanafuda earings. What these heirlooms represent is that they are descendants of the Sun-breathing Hashira. Shinjuro Rengoku goes on to tell Tanjiro (Angrily) that the Sun-breathing is the original breathing style and that all other breathing styles are shabby attempts to copy it or change it.

Basically what Shinjuro Rengoku gets mad at is the loss of his son, but he gets mad at Tanjiro because he believes that Kyojuro died because he wasn't only good enough, but he also implies that the breathing technique wasn't good enough. He believed that Tanjiro was trying to mock him because Ryojuro didn't survive. This brings it around to the fact that Sun-breathing is the original first breathing form that all the other breathing forms branch off from.

If you read the quotes that you had put, you could have been able to understand, but it really does help to watch the show or read the manga. Quoting one of the comments though:

"I think they are the same. It will be revealed/explained better in the upcoming episodes of season 3 which is currently airing" (Fumikage Tokoyami)

To bring it all together and answer your question, no they are not the same, but they are very similar. The flame-breathing form is the closest breathing form to the Sun-breathing form other than Moon-breath which you find out about in the Infinite Palace arc.