What is more apparent in a person's personality? Their rising sign, sun sign or what is the planet that has the most influence in their natal chart?
Could be any or all of those things, and there are other influences, too. If there are any planets in the first house (or conjunct the AC from the twelfth), they get layered onto the ascendant, so the native "looks" a lot like those planets--which might change the flavor of the ascendant very much, if the planets on it are quite different from the sign. Anything aspecting the ascendant, sun, and/or moon throws its influence in there, too.
I think how readily the sun "shines" in the personality has a lot to do with how similar, or not, the ascendant is to the sun sign. If sun and rising sign are the same, the native will probably identify very much with textbook sun sign astrology (unless there are other influences that shade that in a different way). If sun and rising sign are different but share a lot of similarity, ie same modality, there's likely to be easy and apparent manifestation of the sun. But if the sun sign and rising sign are very, very different, that person probably won't "look" very much like their sun. Also, if the sun is placed in a house that's very different from the indicators of its sign, then the native is probably going to look and feel very different from what sun sign astrology will say.
I'm a Leo with a Sagittarius moon, in grand trine with Jupiter in Aries. Read that, and you might think I'm a very outgoing, happy-go-lucky, living large, impulsive type. But I have Capricorn rising. Anyone who knows sun sign astrology only is probably not going to think I'm very Leo--and I didn't think so, either, til I learned more about my birth chart. And my sun is in the eighth house, which has a very different message from Leo. So I'm a case of all of those things manifesting but not necessarily in the way people might expect.
And let's just say a person has lot's of strong influence in their natal chart, then, how would it play out in real life?
If everything in their chart is a focalizer, the native is a focalizer in their society. In some way, they're deeply in tune with the zeitgeist around them. In some way, they have a profound influence on their society. They might become a very famous person, or they might simply be a presence that's felt especially strongly by the people around them.
And let's say the planet that influences your natal chart, does it prevent you from exhibit the traits of your sun sign if the planet is too different?
It wouldn't prevent your sun sign traits from coming out, but if that planet is in a more obvious house than your sun (say, that planet is in your first or tenth house, while your sun is in the fourth, eighth, or twelfth), the other planet's influence will probably be more readily apparent to others, while the sun's influence might be more felt by you than seen by others, or not seen so easily until people get to know you.
However, aspects might bring even the hidden placements closer to the surface. If your sun is, say, in the eighth house and trine your ascendant, it plays an important part in drawing people to you and in how you yourself interact with the world, even though it's in a "hidden" house.