I understand that in BOTH modes, the camera selects the exposure variables, and you just press the shutter to capture the image.
Is the primary difference that on program mode, you can adjust these suggested settings whereas on auto, it is what it is?
Is the answer the same for all makes and models (currently shooting Nikon, btw if it matters).
Read your owner's manual. Most often "P" mode is at the front of the manual section labeled "Creative Modes". Or, as one manual puts it; "More discerning shots, in your preferred shooting style".
In each of the automatic modes, the camera is selecting every value relevant to the end result based on presets created by the designer/manufacturer. Your job is simply to point the lens at something interesting. The metering and focusing systems of the camera then read the overall scene and make corrections based on the assumed "best result" which was assumed on a designer's bench long before you ever purchased the camera. For this reason, most fully automatic shots, whether they turn out the way you had hoped or not, have a very similar appearance one to the next to the next. Sort of like going into a big box retailer, they are all laid out in a similar fashion to provide a sense of sameness to the buyer. You know men's wear will be here and electronics will be there and so on. Or, more appropriately, like eating a fast food meal which you know from prior experience will be like every other meal purchased from that chain. You will not find anything truly unique about the results of either McDonalds or a camera left in full auto mode. However, if at times your idea of creativity is pickles or no pickles, then full auto mode may suit you.
Moving away from full automatic modes to even the simplest of the manual selection modes, the "P" mode, allows you to begin making decisions regarding just how the image might appear. How much control you wish to exert is controlled by which "creative" mode you select. Aperture and shutter priorities allow you, as the photographer, more control over how specific scenes will appear. P mode is still largely an "if this, then that" shooting mode as far as the manner in which the camera's electronics operate but you can at least select the "this". Now you can select pickles but no mustard or mayonaise and extra onions.
You can select multiple values individually in P mode including metering and exposure locks which will provide a less generic look to your photos. These are the most basic functions which determine proper exposure and over look to your photos. Since you select one value and the camera responds but selecting an equivalent value, the camera begins to do as you wish. If you prefer to change the metering of the scene to create an effect, you will not be able to do so in full auto mode (where the camera is in full control of the results) and you'll need to learn how P mode - at the very least - sees things and responds to scenes.
While there's nothing wrong with using fully automatic modes at times, your creative input can only be added in the other than fully auto modes. Certainly, if you have a camera with more sophistication, then learning how to control it vs allowing it to control you will ultimately provide the more sophisticated look of a higher quality image.