Difference between using *.psd and *.png files

No, there isn't any difference.

@AliAzin - Only PVRTC and uncompressed RGB(A) is supported by iOS.

They get re-compressed to PVRTC or converted to uncompressed RGB(A) data when publishing to device. Additionally the psd's layers are not retained when converted to RGB(A).

Note: PVRTC is the default compression for iOS, you will manually have to change it to RGB(A) in the Import Settings Inspector if you want uncompressed image data (ideal for GUI but not Material Textures for 3D objects)

[Edit] I also want to note, for the record, that it is in your best interest to always use uncompressed images to avoid multiple compression degradation. If you use an image that is already compressed (ie a .jpeg) when it compresses it will have degrading effects on the image, kind of like when you save a .jpeg as a copy .jpeg multiple times and compare to the original it is notably lesser in quality as it re-compresses every time.

Cheers

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There is no difference between psd and png because every texture will convert to what you set in inspector. For example RGB(A).