Let's 1st understand what these planets represent on their own?

House Ruler - Besides these karaka things, Sun & Mars will also rule some houses in your chart and will represent things related to that house. Of course we can't cover conjunctions Ascendant and house wise else it will never be over. So, you have to include those things at the time of interpreting this conjunction. Also, we need to keep in mind all Yogas and Raj-Yogas which their conjunction would be making to completely understand the conjunction. Here, we can only cover meaning as per karakas. 

Meaning of Sun-Mars Conjunction - Sun is Father and Mars is Warrior. So, prima-facie indication which I always get with this conjunction is of someone whose Father may be in Army or Police or any other like Services and in 99% cases, it is the case. Even if Father is not in such services, he can be very much warrior like, high on courage and will-power. Person himself can have great will-power. This is a good conjunction to have as both are best friend and Mars is strong enough to deal with fire of Sun. They co-operate with each other well as Soldier (Mars) wants to work with Govt (Sun) and Govt also needs Soldier to guard itself. So, they will mutually co-operate. Sun's light and fire will further increase the courage of Mars and can make person audacious. They always act with tremendous courage. Problem can come if Mars is too near to Sun degree-wise, then it burns the good qualities of Mars. It means, in early life such person can be timid but as he grows up and Mars starts maturing after 28, it becomes again a situation of warrior like person. They are very competitive and warrior like in the area of life where this conjunction falls in chart. That's why, this is not good conjunction for relationship in 7th house as there we need to accommodate with others rather than compete or dominate. They can also be very law abiding and can be working for Govt in high positions. They can also be in Career like Warrior or Fighter but even if they are not, they will always approach their Career with lots of competitiveness and warrior-like attitude. This is a person with very high confidence and ego. 

Who will win the Conjunction? - Here, it can be a balance situation and lots will depend on dignity of planets. If Mars is combust then surely Sun has higher say till 30 years of age, else we have to decide largely through dignity of Sun & Mars. 

Dignity & Strength - 

This conjunction will work best in Aries and Leo as both planets are in good dignity here.

Although, no planet ever likes to be in Scorpio but still Sun-Mars conjunction can do good in Scorpio as it is Mars own sign and Sun's friendly sign but still there will be sudden events/changes and ups & downs in life. Only thing is that person will have strong will power to go through it. 

Sun-Mars conjunction in Capricorn will give higher say to Mars as it is exalted here & Sun is in enemy sign but same conjunction in Cancer will give higher say to Sun as Mars is debilitated but Sun is in friendly sign. Debilitated Mars conjunct Sun person "must" watch for his anger and aggression as it can also lead to violent illegal acts. 

Sun-Mars conjunction in Sagittarius & Pisces will be good for both planets as they both are in friendly sign. 

Sun-Mars conjunction in Taurus & Libra will give more strength to Mars as Sun is in enemy & debilitated sign respectively. 

Sun-Mars conjunction in Gemini & Virgo will be good for Sun as Mars is in enemy sign in both cases. 

Sun-Mars conjunction in Aquarius will hurt both as they both are in enemy sign.