A .vcf file is the format used for vCard, pretty much the standard format for contacts storage.
http://en.m.wikipedia.org/wiki/VCard
If everything is saved to your Google account then everything would have transferred over to the new S5 as soon as you added the account to it and synced.
If you log into your Gmail account on the web, does all your contacts, including pictures, notes, etc you added show up? If not, then you have not saved all contacts to Gmail. You saved some to phone-only storage or maybe the SIM card. Let me try to make this clear...
I've never owned a Samsung, but on many phones contacts can be stored in 3 places.
1. As Google contacts. These are stored on the phone and synced to your Google account (unless sync is turned off).
2. As Phone contacts. These are stored locally on the phone only and not synced.
3. As SIM contacts. Unlike 1+2 above, only names and one number per name can be stored. No addresses, email info, photos, etc.
Your contacts app probably can be set to display any or all of the above. To see what contacts are stored where, go to: Contacts > Menu (3 dots) > Contacts to display > Select account or location (or similar, your Samsung might be a bit different).
If you find that some of your contacts are saved in Phone storage instead of in your Google account, do: Contacts > Menu > Import/export > Export to storage (or similar). Once you have the Phone-stored contacts exported as a .vcf file import it to Google online. Resync contacts to the phone.
Some phones have a conversion function to make non-Google contacts Google contacts so you don't have to do the export/import routine.
Your user manual covers all this in detail.
What carrier the phones are on has nothing to do with any of this.
Android since v1.0. Linux user since 2001.