What is on the roofing side? Asphalt Shingles, roll roofing, etc. When is it due for replacement? You could get a roofing company to add external rigid insulation on top of the tongue and groove, then the roofing. Then on the inside, you could seal any remaining leak of consequence with a clear caulk.
Adding rigid on the inside would certainly help and would allow for effectively air sealing all of those joints. The down side is the thermal short-circuits at the beams. We have done several roofs where we strip the existing asphalt shingles and add several layers of PIC (polyisocyanurate), held down with vertical strapping, and then plywood and typical asphalt roofing details above it, If from the inside, I suggest IKOs Enerair because it is a class III vapor retarder. If adding to the top of the roof, I recommend Dow's Tuff-R. Not all PIC is the same!!
Outside is best as it still maintains the look inside & as mentioned above - no thermal transfer through the beams. With that you do have some issues to consider like soffits, eaves, etc... For more on hot roof options - http://thehtrc.com/2011/what-is-a-hot-roof (see exterior & don't forget to check out the how much needed area)
Is there any insulation or vapour barrier above the wood plank that is visible on the interior?