The DPI value indicates the dots per inch resolution used to print things out. With 300 DPI there are enought dots in one inch to make the pictures look sharp, with 72 DPI artifacts are clearly visible.
But on the Internet, the resolution is irrelevant! Let's make some tests:

Now let's change the resolution to 72 DPI, leaving the physical dimensions at 500x500 px.

What if we change the resolution to 2 DPI? (Don't try to print that out ;)

You can download the files and see that DPI won't make any impact on filesize, it will only affect the printed materials. If you want to reduce the filesize of a picture, you will have to reduce the pixel dimensions.
The font used to “test” is the free Regular Cargo by Nik Thoenen.
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