The GoPro Hero 3 and newer 3 Plus both use the same size batteries and are therefore fully compatible. The older battery is a 3.7V 1050mAh battery and the newer version has slightly more capacity at 3.7V 1180mAh. This ultimately gets more filming time.
However there are a number of good aftermarket batteries that are at least as good as the OEM GoPro versions.
For example the Wasabi GoPro battery is a 3.7V 1200mAh unit. This means that the Wasabi battery has more capacity than either of the OEM GoPro versions.
For just under £25 you can now get a charger plus 2 x Wasabi batteries – Wasabi Power Battery (2-Pack) and Charger for GoPro HD HERO3 and GoPro AHDBT-201, AHDBT-301. Unfortunately the battery charger doesn’t come with a UK plug but does come with a 12V plug for in car use.