as I can see you just edit the main template CSS... you really should use a child template... it's easy to set up and it will avoid any trouble... later...
sorry I'm Franch so the links goes to a translator.. let me know if that help you...
so first of all as you must already probably have , you need a sand box to test
https://translate.google.com/translate?sl=fr&tl=en&js=y&prev=_t&hl=fr&ie=UTF-8&u=http%3A%2F%2Fwww.puce-et-media.com%2Fcr…
then you can use the child template
https://translate.google.com/translate?sl=fr&tl=en&js=y&prev=_t&hl=fr&ie=UTF-8&u=http%3A%2F%2Fwww.puce-et-media.com%2Fcr…
depending on the thme and the needs you probably will need to manage dependencies... so have a look here
https://translate.google.com/translate?hl=fr&sl=fr&tl=en&u=http%3A%2F%2Fwww.puce-et-media.com%2Fchild-theme-gerer-les-de…
if you need to go further with your child template... further than CSS
https://translate.google.com/translate?hl=fr&sl=fr&tl=en&u=http%3A%2F%2Fwww.puce-et-media.com%2Faller-plus-loin-avec-un-…
then once everything is done in the sandbox, the major problem become to transfert all these data (made in WP, not in the child template)... as menu, trees, articles, portfolio... in fact local DataBase... so
https://translate.google.com/translate?hl=fr&sl=fr&tl=en&u=http%3A%2F%2Fwww.puce-et-media.com%2Ftransferer-son-site-loca…
let me know if that helps you... because to answer to your initila question, if everything is set correctly (in DW website setting) and in your local computer stack localhost (Xampp, mamp, easyPHP, wamp, ... lamp...) there shouldn't be any difference...