For some reason everyone overcomplicates this.
- Android 5.0 or 5.1
- Minecraft PE v0.14.1
td/dr version: Connect both tablets to your PC as storage devices, then copy the following folder from the old tablet to the new tablet: \games\com.mojang\minecraftWorlds
Hint: Minecraft seems sensitive regarding the version of the game. I suggest that both tablets have the same version of Minecraft installed before you try the transfer. I don't know if this is required, though.