Why your Sun sign changes between Western and Vedic
Two systems, one sky, and a 24 degree gap that has been widening for two thousand years. Neither of them is wrong.
If you have ever been told you are a Capricorn and then read somewhere that you are a Sagittarius, nobody made a mistake. You were measured against two different starting points.
The two zodiacs
Western astrology uses the tropical zodiac. Zero degrees Aries is defined as the spring equinox, the moment the Sun crosses the celestial equator going north. It is a seasonal marker, tied to the relationship between the Earth and the Sun rather than to any star.
Vedic astrology uses the sidereal zodiac. Zero degrees Aries is defined against the fixed stars, the actual constellations. It is a stellar marker, tied to the background sky.
Those two starting points used to coincide. They no longer do, because the Earth wobbles.
Precession, and the gap it opened
The Earth's axis traces a slow circle, one full turn roughly every 25,800 years. That means the equinox point drifts backwards against the stars by about one degree every 72 years. Around the third or fourth century the two zodiacs lined up. Since then they have separated by roughly 24 degrees.
That gap has a name: the ayanamsa. Crescent uses the Lahiri ayanamsa, which is the Indian government standard and the most widely used. Because a sign is 30 degrees wide and the gap is about 24, most people's placements move back by one sign when you switch systems, and the ones near a boundary do not move at all.
So which one is right
That question assumes they are answering the same question. They are not.
The tropical zodiac tracks the seasonal year, so it is describing your relationship to a cycle of light and dark, growth and decay. The sidereal zodiac tracks the actual stars, so it is describing your position against a fixed backdrop. A Western chart tends to be read psychologically, as character and pattern. A Vedic chart is read predictively, with the dasha system supplying timing that Western astrology traditionally handles differently.
Crescent shows you both because the disagreement is informative. When both systems put a planet in the same house, that placement is doing something unambiguous. When they disagree, you are looking at the difference between how you experience something and how it plays out.
What to do with it
Read the Vedic chart for timing and the Western one for texture, and do not try to average them. If you only want one, pick the tradition whose readings have actually described your life, not the one whose Sun sign you prefer.