I know that we get all excited about having digital cameras, but the fact is that we don't. We have analog cameras that happen to have a digital output format (and a lot of digital circuitry that is not directly related to the image as well).

Whenever an analog signal is converted to a digital signal, you introduce quantization noise. That is, it's very unlikely that the signal going into the conversion will exactly match the value of the digital number that comes out the other end — think of it as truncation or rounding errors.

When you do post-processing on a digital image file, the quantization noise the camera added is "baked in". It doesn't matter what bit depth you're working with in post, the data that you are working with has both the analog (well, quantum) noise components (thermal and shot noise, amplifier distortion, etc.) and the quantization noise of the camera's output. The quality of the base data is not perfect, so any computation done on the faulty data will result in faulty output. GIGO, as they say.

In-camera, on the other hand, you get the opportunity to amplify (or attenuate) the analog signal before quantization. That doesn't help at all with noise in the analog domain, but it does reduce the quantization noise at a given brightness level.

Let's say that you have an analog value of 4.4 whatchamacallits. If you shoot using ISO 100, our hypothetical "digital" camera will convert that to a digital value of exactly 4. If you choose to increase the apparent exposure in post, you're stuck working with the 4, which is truncated. If you increase the ISO in the camera (by less than a full stop), that 4.4 will be amplified by analog circuitry before it's converted to digital, and may result in a 1-higher digital value than the all-digital processing computes. A single-bit difference may not sound like much, but when you start accumulating all of the errors along the way in processing, a given pixel might be quite a long way from the values it ought to have. That's what noise is.

(There is also the fact that the camera "knows" its own response characteristics, and can account for them in processing. Lightroom, for instance, doesn't do camera-specific, ISO-based sensor noise subtraction. Cameras can, though not all do.)