Active Directory Administrative Center over Active Directory Users and Computers

Is there any difference using Active Directory Administrative Center over Active Directory Users and Computers?

Yes, ADUC is a subset of ADAC. ADAC is newer and allows for more complete administration. For example, you can enable the AD Recycle Bin, recover deleted objects with the Recycle Bin, and create a Fine Grained Password Policy from ADAC, but you can't do any of this from ADUC.

In training videos I've seen people mostly use the older look Active Directory Users and Computers and was wondering if it really matters?

For day-to-day administration (resetting passwords, managing group membership) probably not. You can do that from both.

Also, I've seen people add Users with Active Directory Users and Computers and they show in same window with the Groups. However, if you use Active Directory Administrative Center to add a user, you won't find the users in Active Directory Users and Computers unless you open up the group that you added the user(s) too.

I have no idea what you're trying to say here.

Is there a correct approach to working with users, groups and computers in active directory as far as which style is used?

ADUC is still very widely used. It's a familiar interface that's been around for 15 years now. ADAC hasn't been widely adopted (yet), but contains more features. If you want to be ahead of the curve and you're just learning now, I'd use ADAC first.