The zodiac sign generally refers to just the sign (Aries, Taurus, etc) of the Sun's position in your natal chart. In astrology the interpretation of the Sun's position in your chart will be determined by the combination of the sign, what's called the "house" it's in, and the exact degree of its placement (sometimes called the "sabian symbol").
But more complicated than that, you also have the placement of all the other planets in your chart to consider, and how all of those relate to each other. You can understand aspects of your planetary placements in isolation but all the intricacies of it can't be fully grasped that way as all planetary placements are in relationship and that effects the interpretation of each of them.
The best metaphor I can think of to explain astrology is like a "chord" in music. Each of the signs, the planets, the houses, are all like notes which have a distinct tone in isolation. These notes combine together to make the unique "chord" of your planetary placement. As such "Aries - Sun" for me could sound a bit different to "Aries - Sun" for you even though they will share common notes. The unique "chord" of all your planetary positions's chords being played in unison makes up the unique sound of your natal chart and you as a person. In theory, according to astrology, you're a unique mirror and personification of the unique "sound" of the moment you were born.
This means there are so many variables in a natal chart that there are virtually endless combinations. Every natal chart is unique, just as every moment and every person is unique and never an exact clone. As a result, astrology as a system is too complex to understand with ease in a reductionistic way and is why interpretation is inclined towards accessing intuition and reading it holistically in terms of patterns.
It's because of this level of specificity -- and from what I've found for myself at least, accuracy -- that I take personal interest in it as a more intricate framework for modelling a person than others like personality types or MBTI I've been interested in before. To me, reading someone on the basis of their personality type is overly simplistic in a similar way to reading someone in terms of their zodiac (sun) sign (which is what horoscopes are based on). Seeing your type as a mirror of you is similar to seeing your horoscope as a mirror of you. It might have some accurate information but I'm unlikely to trust the overall picture as there's a lot both don't take into consideration that distorts the image. I'd imagine that people are attracted to either of these things for similar reasons.
I know people here are likely to be skeptical of everything in this post so it's more a way of explanation than persuasion because I happen to be bored at work. If you don't get much out of your natal chart on first glance, I'd suggest that reading a natal chart takes a fair amount of investigation and is to be done for fun because you find it fun. Also, like another person said, it's worth double-checking you have your accurate birth information as that can throw off a few important variables.
Peace xxx