If you've only ever encountered Western numerology, "birth number" and "destiny number" both sound like they should mean the life path number — the one calculation everyone eventually lands on. Indian numerology, though, tends to keep the two separate on purpose, treating the day of birth and the full birth date as two different sources of information.
The birth number is the smaller of the two jobs. Look at the calendar day you were born — nothing else — and reduce it. Someone born on the 4th, the 13th, the 22nd, or the 31st all end up in the same place: a birth number of 4.
The destiny number widens the lens to the entire date. Day, month, and year all get summed together, digit by digit, and reduced the same way. It's a heavier calculation working with more raw material, and in most schools it's treated as describing something broader — a life direction rather than a surface trait.
