If I have created a separate Photoshop and Element version of a product then, the Elements version of all the templates can be used in Photoshop and Elements. However, the Photoshop version cannot be used in Elements.
Photoshop is a more robust program offering more features and layer FX. Also Photoshop gives the option to organize layers in folders. Since Elements does not have the option for Folders, when you open the a template with Folders in Elements you cannot get to the layers in the folders and that's what causes most of the problems. In the case where the description says it will work with Photoshop or Elements, that usually means that it has no Folders or special FX that Elements cannot handle.
Photoshop also offers more advanced Smart Objects which can be used to easily place images in some of our templates.
In conclusion, if you want to use the templates in Photoshop or Elements then I recommend that you download the Elements version of the template.
If you upgrade to Photoshop, you can go back into the download area and download the Photoshop Version. Both are available to you for your purchase.
Other resources for the difference between Photoshop and Elements: