How to differentiate the homepage structure from the category page structure in WordPress template?

If I understand you…

Your theme may only have an index.php and a single.php but WordPress supports a much more complicated templating system, and you can control the formatting of different types of content just by creating files with the right names. In your case, I think that at least one of the files you need is category.php.

Of course, you will have to write the appropriate PHP for that template but you should be able to copy either your index.php or your single.php to get started.