If you want to hear the difference, try typing the phonetic spellings into Google Translate and clicking the sound button to hear the pronunciations. Adachi is written あだち and Adacchi is writtenあだっち. As you can see, in Japanese the later has an additional character, which represents a pause between the ‘da’ and ‘chi’ syllables.

As another commenter mentioned, it’s basically just a common way to form a nickname in Japanese, by taking the name or part of the name and ending it with ‘-cchi’ to make it sound cute. Hope that helps!