...... the sign and house of the Sun color the whole chart and show where the maximum focus will be had. (Mars will never be equal, IMO). The Sun is VITAL. Vitality is the keyword for the Sun. Whatever the position of the Sun shows will be vital in more than one way for the querent. Always, as long as the querent is alive........ You must always take the sign of the Sun as well, IMO. You must first decipher the "personality" of this Sun to then proceed and see how such personality would act in the area of the house. A Libra Sun in the 5th will be interested in things different than a Virgo Sun in the 5th.
I wouldn't disagree with the general nature of Ronia's comments, though I'm going to add a few other points. I would start with echoing Ronia and adding one other planet. Through the history of Horoscopic Astrology (the Astrology that uses a chart based on the Ascendant), the Sun and the Moon have always been two of the three most important planets in any chart. The other planet is the Ascendant Ruler (and the Ascendant itself is the most important point). That means that for 100% of us the Sun and Moon are of great importance for some, with Leo or Cancer rising, they will be paramount. However, for people with Aries or Scorpio rising, it's possible that Mars will be equal (on average that will be one sixth of births) in terms of determining life experience. For non-Aries/Scorpio Ascendants, Mars may be of little influence at all.
Ronia says that the Sun represents vitality, nearly 2000 years ago the Astrologer Vettius Valens gave the characteristics of the Sun as
'
bringing light to the mind'....'signifying that part within us that is inwardly noble....that part that can take on leadership responsibilities....the part of the mind that is employed in perceiving, thinking, feeling, deciding...the ability to be wise and know things from experience'
I think the word 'VITALITY' is a good choice of a single word to sum up those characteristics. Take away those characteristics and life becomes aimless, meaningless and chaotic.
It is no wonder that the Sun is important but the Sun's characteristics are not all that allows life to be maintained. The Moon according to Valens
signifies the internal property of "being"...the quality of "existence"....connected with the physical body.
He goes on to link the Moon with the need for companionship, marriage, nurturing, the facade or mask we wear in public (often attributed to the Ascendant by modern astrologers) and the "goddess energy within us", By medieval times the Moon was seen as the instinctive, non-rational part of our minds.
So the Sun alone is not enough, the Moon is also necessary to sustain life. The two used to be referred to as the Lights. Which of the two is seen as the most important has varied over the lifetime of Astrology. Modern Astrology takes the Sun but it has not always been so. At the beginning the most important of the two was the Sect Light. That is the Sun for day births (Sun above the horizon) and the Moon for night births (Sun below the horizon). By Medieval times, the Moon was seen as being more important as it had rulership over things below the heavens, whereas the Sun had rulership for things above. However they were still the two most important planets, in general.
The third planet, which differs from birth to birth, is the Ascendant Ruler. That planet says something about our mind, body and spirit. It represents our physical body, ourselves. For nearly two thousand years these three have been the triumvirate of most important planets in any chart.
Ronia is right to draw attention to the Sign placement. The character of each of the three is nuanced for better or worse by the Sign in which it lies. A medieval view was that the nature of the Sun cycles throughout the year, according to the seasons, for Autumn births it adds a touch of Melancholy, for Summer births it adds a touch of Choler, or Fire.
Returning to the original question, it depends on what you mean by 'important' or perhaps better phrased as 'important to whom?' The more likely answer is that the Ascendant Ruler by house gives the area of life that is most important from an objective stance.
I stress 'objective' because the person themselves might not see things that way. I'm talking more about how others see them.
There is also the issue of what house system is being used because it is quite possible for the Sun and indeed any planet to change houses if a different system is used. That doesn't necessarily invalidate the point. Quite often a person might be seen differently by different people or groups.