Why do Western Astrologers say the New Moon is in a different sign?

Vedic Astrology uses the sidereal zodiac which is astronomically correct. You simply locate the planet in the sky and the constellation behind it, is the sign.

Simple.

In Western astrology, the Tropical zodiac is used, linking Aries to the Vernal Equinox. This lined up astronomically 2000 years ago, but due to the earth's wobble (precession of the Equinox), it is now 24 degrees off and astronomically incorrect. I highly recommend an app on your smart phone like Skyview or Sky Guide so you can test this out.