They are NOT different! Someone might say that “Physics is a branch of Math that studies how this world works”. And someone else might say that “Physics is a science that deals with matter and energy and their interactions, and Mathematics is the language of physics”. Or as expressed by Galileo Galilei: "The book of nature is written in the language of mathematics". And since physicists study nature they need to understand math.
On the other hand, there is no generally accepted definition for mathematics. Mathematicians seek patterns and use logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. A great many professional mathematicians take no interest in a definition of mathematics or consider it undefinable. There is not even consensus on whether mathematics is an art or a science. Some just say, "Mathematics is what mathematicians do."
And it is worth mentioning that a lot of mathematicians have contributed to physics and a lot of physicists have contributed to math.
Sir Isaac Newton for example was an English mathematician & a physicist. His book was titled “Philosophiæ Naturalis Principia Mathematica” or Mathematical Principles of Natural Philosophy.
Carl Friedrich Gauss was a German mathematician & a physicist. Leonhard Euler was a Swiss mathematician & a physicist. and the list goes on...
Maybe from a narrow preceptive someone might see these fields of knowledge as unrelated, but as you learn more and look at the big picture you will see the deep connections between these subjects.
Lastly, here are two “maps” of these huge subjects created by Dominic Walliman.
The Map of Physics: https://flic.kr/p/Pcj581
The Map of Mathematics: https://flic.kr/p/Ra6UW5