Both fp16 and bf16 use 16 bits total — half the size of ordinary fp32 — but split those bits differently.
fp16 (IEEE half precision): 1 sign + 5 exponent + 10 mantissa bits. More precision, a narrower range — values that are too large or too small simply overflow to infinity or zero.
bf16 ("brain float", from Google): 1 sign + 8 exponent + 7 mantissa bits. The same exponent range as full 32-bit floats, less precision — far more forgiving during the unstable arithmetic of training a neural network.
Same bit budget. Different trade-off between "how exact" and "how big."