You are asked for the recovery key after you opted to reset your password (ie. on myteamspeak.com).
All synchronized items are encrypted with your account password, before the client sends them off.
When you reset your password, the password does no longer match that with which the items have been encrypted and as such they cannot be decrypted. That’s when you need your recovery key that was provided to you.
If you use the same myTeamSpeak account on both devices, then both will ask for a recovery key.
In TS3 there will be an exclamation mark icon in the bottom center on blue square background (at least in the default iconpack). If that icon is displayed it means that the client cannot decrypt your synchronized items and cannot request a new recovery key. You can double click or right click said icon to have the option to enter your recovery key. If it gets rejected try again a few times.
Please note that when using the new client, you will most likely have received a new recovery key when you first logged in, which is the only valid one for the account from that point on.
If you don’t have your account recovery key, the only other option is to use the fallback mechanism. This will erase all your synchronized items on our systems. The client you’re using the fallback mechanism on will then try to upload its locally cached items if there are any.
You can request a new recovery key in Tools → Options → myTeamSpeak → New Recovery key, however that only works if the client can currently access your data (that is if your account is not in recovery mode, ie. asking for a recovery key to be applied).
