Ignoring small craft and ships operated by minor navies, the last losses of surface ships at sea to weather occurred during WW2. During the war, Italy, the Soviet Union and the USA all lost destroyers to weather. A French ship broke in half in a moderate gale, but this was mainly due to high stresses in the upper deck intrinsic to the design. The majority of the American losses occurred after Admiral Halsey took his Task Force 38 into a typhoon in 1945. The losses to weather during the war are shown in this table:

Ship	Country	Date	Cause of sinking
Branlebas	France	14 December 1940	Broke in half
Lanciere	Italy	23 March 1942	Lost power and foundered
Scirocco	Italy	23 March 1942	Unknown
Sokrushitelnyi	USSR	22 November 1942	Pooped, lost power and broke in half
Warrington	USA	13 September 1944	Lost power and foundered
Hull	USA	18 December 1944	Capsized
Monaghan	USA	18 December 1944	Capsized
Spence	USA	18 December 1944	Capsized

The three USN destroyers sunk on the 18th December 1944 would ordinarily have survived the storm that sank them. However, they were low on fuel, and had not taken on ballast. This meant that they were riding higher in the water than expected and were hence less stable. Also notable is the loss of the American submarine USS Cochino on the 25th August 1949. She was so rocked by the heavy seas that her batteries started to leak. This caused a chemical reaction which produced hydrogen gas. This exploded, sinking the submarine. On the 6th February, 1968, the WW2-era destroyer USS Bache, moored in Rhodes harbour, dragged her anchors in hurricane force winds. She was blown onto rocks, and was so heavily damaged that she was not salvaged.

As to why these losses stopped post-war, it's a combination of new standards, better simulations and better weather forecasting. The stability of a ship at a particular angle of heel is determined by its 'righting lever', or 'GZ', which is the distance between its weight and buoyancy vectors. By plotting this against angle of heel a 'GZ curve' can be obtained. This shows how likely the ship is to right itself, at a particular angle of heel - if the ship's maximum 'GZ' is after the point where its intakes start to flood, then it will never right itself, and will capsize. The USN introduced new standards for ship stability after the loss of Hull, Monaghan and Spence, which would later be copied by other NATO members. These standards required that the wind force on a ship should be equated to the righting force from the GZ curve, with a margin of safety for rolling. Intakes were better protected from spray and rain, preventing water from entering the engines and causing ships to lose power. Computer-aided design, and the experience of the war allowed for better calculations of stress and strain within ships. This eliminated several weak points within designs. For example, most WW2-era RN ships had a break at the end of the forecastle, which acted as a stress concentrator. Several ships developed cracks at this point, and had to have strengthening added. This was eliminated in post-war designs, removing this vulnerability. Better weather forecasting has an obvious effect, allowing ships to avoid bad weather. Halsey's steering Task Force 38 into Typhoon Cobra occurred partly because forecasters at the time couldn't predict where the typhoon was going. Now, Halsey would be able to avoid the typhoon, or take enough precautions to ensure that the task force took minimal damage.

Sources:

Nelson to Vanguard: Warship design and development 1923-1945, D. K. Brown, Seaforth, 2012

Rebuilding the Royal Navy: Warship Design Since 1945, David K Brown, George Moore, Seaforth 2012

Three US destroyers were lost in Typhoon Cobra in 1944. A carrier was also very hard hit--the Monterey. One of those affected by the fire breaking out due to planes crashing around went overboard. He also went on to become President. That man was Gerald Ford.