Looking at the correct textpad version of the file, it looks as though the tab size here is 8, since the space between the number and the first column is eight characters. Perhaps TextPad has an option to autodetect tabsize. Sublime will do this too, but it’s autodetection routine is designed more for code than arbitrary plaintext files.
Regarding tabs, Sublime is only different to most other editors in one limitation, that being that tabstop positions cannot be set independently. So in Sublime if you set tab size to X, there will be a tabstop at every X position. Some editors allow you to set tabstops like electric typewriters, say, stop one at 10, stop 2 at 30, stop 3 at 50, then stops at every 5 after that so 55, 60, 65, 70 etc… When routinely dealing with tab delimited text that can be very useful and is lacking in Sublime unfortunately. In such cases I usually fall back to another editor (Crisp in my case), though of course I’d love to see this added.
Sublime can operate in “tab” or “spaces” mode (indicated bottom right in the status). The mode simply determines what the tab key and autoindent will do; in spaces mode it will simply insert as many space characters as are needed to get to the required tabstop. Regardless of mode, Sublime won’t convert tab chars to spaces and vice-versa automatically on load or save, but you can do this manually (View…Indentation…Convert to tabs/spaces).