In short
Human Design and the natal chart start from the same calculation: from the moment of birth and the ephemeris, they find where the Sun, Moon and planets stood. After that, astrology sorts those degrees into zodiac signs and houses, while Human Design sorts them onto a wheel of 64 Gates and adds a second snapshot, taken 88 degrees of solar arc before birth. The natal chart describes character in the language of qualities; Human Design gives the mechanics of decisions: wait for a response, for an invitation, or inform.
A common root
The natal chart and the bodygraph begin the same way: take the moment of birth, convert it to universal time, and use the ephemeris to calculate where the Sun, Moon and planets stood at that instant. Up to this point the systems are indistinguishable — the calculation is literally the same.
They part ways at the next step. Astrology sorts those degrees into zodiac signs and houses. Human Design lays the same degrees over a wheel of 64 Gates taken from the I Ching, and adds a second snapshot — 88 degrees of solar arc before birth.
Different questions
The natal chart answers the question “who am I”. It describes character, inclinations, areas of life, your relationship with money and people — and it does so in detail, in the language of qualities.
Human Design answers the question “how should I act”. It says almost nothing about character, but it gives the mechanics of decision-making: wait for a response, an invitation, or inform; decide with the body, with time, or with your voice.
So there's nothing for them to argue about. Astrology describes who you are. Human Design describes the way for you to use it.
What to take from each
If you want to understand yourself and the people close to you, take the natal chart: it has more language for describing what you already feel.
If the question that torments you is “why do I keep doing the wrong thing and getting tired”, take Human Design: it's about where you push life by force instead of waiting.
And Vedic astrology — Jyotish — adds a third view: it describes a person by the Moon rather than the Sun, and so it speaks not about intention but about how a person actually lives and feels.