Why is PDF much larger than PNG for a vector design?

Following PDF only with JPG photo is 7 times larger than original one solution could be to use JPEG compression for raster images instead of the default ZIP compression, which is similar to PNG. At JPEG compression of maximum quality, the PDF file weighs 1.9 MB.

In general

Let me spam you another answer: Svg file size more than a png image

Here is a graph where the x-axis is the pixel size and the y-axis is the file size.

A vector file is mostly independent of the dimensions and a raster image is proportional to it.

But that is not the full story.

Now let us think in an uncompressed raster image vs a vector file both with fixed dimensions. But now let's change the x-axis, now it is "information".

You can have one tiny or several circles defined in the image. On a bitmap the diference will be the color of the pixels but the pixel count is the same. The size of the file will not change.

But now put two circles on a vector file vs one. You have doubled your information. Let us think something like:

<circle cx="50" cy="50" r="50" stroke="black" stroke-width="5" fill="black" />

Now put 3, 4, N circles.

<circle cx="50" cy="50" r="50" stroke="black" stroke-width="5" fill="black" />
<circle cx="50" cy="50" r="50" stroke="black" stroke-width="5" fill="black" />
<circle cx="50" cy="50" r="50" stroke="black" stroke-width="5" fill="black" />
<circle cx="50" cy="50" r="50" stroke="black" stroke-width="5" fill="black" />

The increase in the information is linear. The information will increase for any new feature added to your file.

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On your specific image

You have a shadow. This can be an actual bitmap for faster processing or could be the instructions on how to re-render it each time. This can make a difference in file size, but also in processing time. Imagine you need to repeat the instructions each time.

On your update, you mention linking an external photo to a file. On a PDF file, there are not linked files. They are inside the file. The file inside could be resampled, and compressed, or not depending on the settings. It can also have fonts converted as curves or the font file embedded or not.

On a native file you have the file actually linked, but with a preview embedded.

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Conclusion

There are a lot of factors to consider thinking about a file "weight". Vector information, instructions, raster effects, raster images, preview, scripts, layers, embedded information, objects not displayed, etc. You can not think about file size just in terms of vector vs raster.

You can not actually know how big the file is going to be beforehand. It is perfectly possible that a PNG export is smaller than a PDF file this is especially likely if your file has big raster elements in it since the PDF will contain the full raster data + all layering of such raster data + all the other data while a PNG just contains one flat layer of pixels.

Now its impossible to know how big a PDF files vector data is going to be to do so i would need to see the vector construction and cogitate on it. But lets just say since you have a font on the page and about half of what is on the page is vectors its not entirely unreasonable that this would be half the file. If it would be tripple the size of the file or just half wouldn't supprise me to the slightest.

Since you haven't shown your PDF export settings, I can only make some guesses.

If you have the "Preserve Illustrator editing capabilities" option checked when exporting the PDF, that could explain a larger than expected file size. Basically, that setting saves both the AI file data and the PDF data in the one file. Try disabling it. Obviously, make sure that you've saved your AI file first, since the PDF will no longer be properly editable in Illustrator.

Effects such as drop shadows will be rasterized when exporting to PDF. If your PDF settings are set up for high quality printing, this means you could have one or more large raster images contained within the PDF. PDFs can contain both raster and vector images.

Also there's no guarantee that a pure vector image will be close to a PNG's file size. Vector file sizes depend on how many paths, anchors and individual objects there are in the design (i.e. the amount of data), and the file size of a PNG will depend on how many pixels are in it, and the colour mode (i.e. indexed versus RGB, or RGBa). Vector and raster file sizes are not really comparable.