Is there any practical difference between saving at 300ppi or 72ppi?

There's no difference. The statements you are seeing are uninformed.

There are a lot of uninformed people on the internet, so that's not surprising. You haven't missed anything, except maybe you are overestimating the reasonableness of typical comments you might find online. :)

I suppose that it depends on who is doing the speaking.

If the speaker has half a clue, you should understand their statement to mean that they're saving the image at the same dimensions but lower resolution. That is, if your goal is to provide a preview image that can't be used to make high quality prints, you save it as (for example) an 8"x10" at 72dpi instead of as an 8"x10" at 300dpi.

If the image that the speaker is saving turns out to be 24"x40" at 72dpi instead of 8"x10" at 300 dpi, you can assume the speaker has a fundamental misunderstanding of how images work.

Does an image edited and saved/exported with 1200x800 pixels at 300 ppi look any different online than the same image saved/exported with 1200x800 pixels at 72 ppi?

No.

A bitmap produced either on-screen or on paper from the image will be identical.

The only difference would be the default print size from some applications, and only then if the image size is not specified in any other way.

If you were to open the image in a simple image viewer (something akin to MS Paint that handles images, but not page layout, such as Adobe Illustrator) and press Print, you may find it sets the default print size based on the resolution, so the 300 PPI print would be 6x4" while the 72 PPI print would be 17x11"... assuming it didn't just auto-fit to the default paper size.

The only way to stop someone printing it at higher than 72 PPI on a 6x4" photo would be to only upload the image at 432x288 resolution.

As you wrote it, the answer is that there is no difference (until you print it or look at it in a document that will be printed).

First a clarification: PPI is pixels per inch, a description of the resolution of the image. DPI is dots per inch, a description of the physical ability of the printer/scanner being used.

pixels (on a side) = ppi x inches. multiply the two sides to get the total size of the picture, usually measured in MP (megapixels).

My local paper runs a section where people can submit their photos, and the instructions are that the pictures "must be at least 300 PPI". I've always been tempted to turn in a 300x300 pixel image at 300 PPI. This of course is low-res (.09 MP) and only prints at 1"x1", but would meet their posted requirements.

The assumption behind statements like this is that the image is a "reasonable size" when printed (like 4"x6" for example). In that case, 300 PPI means it would also look sharp (contain a lot of pixels, in this case 1200x1800=2.2MP), compared to 72 PPI which would look, well, pixelated at .1MP.

In your example they are measuring by pixels ("1200 pixels on the long edge"), not in inches, so then the PPI value is immaterial.

As a professional photographer for 30 plus years I can assure you there are considerable differences between DPI values. It might be better to consider DPI as a factor about your final intentions for your photograph. As some choices are not reversible easily it is often sensible to keep two or more versions of your image . A lower 72 DPI version will display perfectly well on any social media and most pc's and TV's. However if you intend to make a print or take your image to your local colour lab then you will need a 300DPI version to use as the master copy to ensure you get the best result. The reason photographers choose raw format generally is to maximise the information in the image against the compression technology that occurs using JPEG format. Also you need to remember that even with the leaps and bounds of the new Ai software's for upscaling the simple rules are that you can easily down size your 300DPI image to 72 DPI without any problems , however if you try to bring your 72 DPI image up to 300 DPI you will loose a considerable amount of detail permanently . Hence you need to keep your 300DPI image as the master version and make all your other versions from it. There are many monitor screens running at 96 DPI these days so most pros save at 100DPI for social media and internet work .

For completeness, I'll add an example when the PPI tag matters even without printing.

Some modern programming environments/platforms (for example, .NET WPF uses device independent pixels) do render images on the screen according to their PPI/DPI setting, rather than pixel-to-pixel as was always the norm. This is done in order to accommodate higher resolution displays, which are becoming more and more common.

If you want your image to be displayed pixel-to-pixel at least on normal screens (96 DPI in Windows), you need to save it with 96 DPI. (And even then do some tricks which are beyond the scope here). This still catches many a developer by surprise, who have a habit of always thinking in terms of pixels.

Then, like in the "real" world, using a high resolution display will not benefit this image and may even make it a bit worse, because it will be automatically upscaled according to the OS settings. On the other hand, one can realise these benefits by saving a 200-300 DPI image and rely on downscaling on "normal" displays.

This, of course, is mostly relevant for "graphics" in practice, but equally applies to photographs if they are used in the GUI.