Templates share their content. You'll have to clone the Template (Properties->Template->Save as new template). This creates an ad hoc (=unique)template based on the original one. If you edit your source page template, there's an option to always create ad-hoc templates from this source when creating a new page.
This is very useful for content templates. E.g. I have a page template "Default Content Page" with that option turned on. Everytime I create a new page based on that template, Kentico will automatically create a clone of my source template.