I do not understand what/where the problem is. I tried using Illustrator too, but it still exported with a white background. Transparency was checked every time in both applications.
As said in the other answers, no export setting can create transparency where there is none. You can, however, use a quick trick to remove a white background form an image like this is using
- Open your image in Photoshop.
- Make sure the image is a single, regular layer, not a background layer. If it is a background layer, double-click its bar in the Layers palette and choose 'OK' to make it into a regular layer.
- Double-click the layer's bar in the Layers palette again so the 'Layer Styles' window appears.
- At the bottom of that window, find the 'Blend If' settings. The Setting after the words 'Blend If' should be set to 'Gray'.
- Focus on the top bar, that says 'This Layer' and find the gray arrow at the right end of the bar.
- Press and hold Alt / Option and drag the left half of that arrow slightly to the left. In your example image, I got nice results with a value around 220.
- Click 'OK'.
- Export as a .png as you are used to, ensuring to check the 'transparency' box.
The checkbox will not magically transform a white background into a transparent one.
In the screen capture, you posted the background is clearly white. If it were transparent you would see the gray checkerboard.
See if you have a white layer that needs to be turned off before exporting, or create a mask.
In Photoshop you will see that images are transparent if the background is marked as a cheqerboard.
Here you can see the non-transparent on the left, and a transparent on the right.
If the PSD file you downloaded have layers, you might me able to "make it transparent" by disabling the (white) background layer before exporting to PNG. Access the layers from Window -> Layers.