I created a forum so that I could work on a skin to use on a forum that I'm a moderator of. On the forum that I created, I noticed that the profile section looked entirely different to what I'm used to.

What I'm used to:
https://2img.net/h/oi28.tinypic.com/aexlyv.png

What the profile looked like:
https://2img.net/h/oi27.tinypic.com/9tpo9u.png

Can anybody tell me how to update the forum that I am currently moderator of (so that I may pass it on to one of the admins)?

Thank you

Seems like one is Invision and the other is PHPBB(?) so you will need to check the exact theme of the forum.

No, I know that they're both phpbb2...I checked before I made the second forum

Um i believe that if you simply just change the theme of your forum (because i tested this as well) it will show up because the css sheet on the forum your on probably was given the updates to the profile page so thats why it looks the same.

How do I change the theme? The skin of both forums is exactly the same.

Your problem has got nothing to do with skin issues.
You can change it via admin panel
Users & groups>Profiles
General Options tab

Activate advanced profile :
If you choose yes : https://2img.net/h/oi27.tinypic.com/9tpo9u.png
If you choose no: https://2img.net/h/oi28.tinypic.com/aexlyv.png

Hope that helps.