The painting in the museum is J.M.W. Turner's The Fighting Temeraire
This is a painting of a ship that had played an important part in the Battle of Trafalgar, but 34 years later, was being towed to be broken up for scrap.
(File from Wikipedia Commons)
That bit was easy to work out. It is one of the most famous British paintings and I recognized it immediately.
The painting from M's office is much less well known, but is picture of the Battle of Trafalgar, showing with HMS Victory with other ships. The clever part from the Writers/Director of Skyfall is that the Temeraire was the ship that immediatelyclosely followed HMS Victory into battle at Trafalgar where the British won convincingly against a larger force, the Temeraire capturing two "prize" ships.
The suggestion from the appearance of this pair of paintings is that we are no longer seeing a tired old warrior any more, but same warrior is seen as if reborn, ready for action.
It It is clearly no coincidence, and the director is using the pictures to symbolize Bond rejuvenated and ready to carry out the next mission for Her Majesty's Government.
There is a more detailed analysis in this excellent blog post. Great Question, made me look into this.