What you're seeing is the effect of the US Coinage Acts of 1853 and 1873, and the Bland Allison Act of 1878.
The silver dollar was based on the weight established by law under the Coinage Act of 1837. Under the act, coin sizes are based on assumed weight ratio of 16:1 (i.e. 16 oz silver is 1 oz gold). The weight of a silver dollar was set to be 412.5 grains (26.73 g) of 90% silver. A quarter was thus 103.125 grains.
The risk of this system was that if the global price of silver increased relative to gold, silver coinage would flow out of the country because would be worth more melted and sold on the global market than as coinage. Gold rushes in California and Australia significantly increased the global supply of gold, creating precisely this scenario.
In response to the disappearance of silver coinage, Congress thus decided to change the size of the silver coins (5c to 50c) in 1853. But the silver dollar was left unchanged: the size change was a intended to be temporary measure, and it was felt important to send the signal that the currency still had inherent value through its silver content. In practice, the change was permanent and the silver dollars basically disappeared.
There was another slight change to the sizes of the coins again in 1873, and the silver dollar was discontinued at the same time. After the 1873 Act, a quarter weighed 96.45 grains or 6.25 grams.
The 1873 Act - more controversially - also removed the right of holders of silver bullion to have their silver turned to coins. This made it possible for silver coinage to have an intrinsic value lower than face value, pushing the US away from bimetallism and toward the gold standard. The Morgan Dollar was authorised five years later largely due to opposition to this policy: the Bland-Allison Act required that the government buy large amounts of silver to turn into silver dollars at the 1837 16:1 weight ratio.
The British history is less involved. From 1816, the weight of British silver coins was based on the assumption that 1 troy pound of sterling silver (92.5%) was worth 66 shillings. As a troy pound is about 373.24 grams, that gives you 5.66 grams for the shilling and 28.28 grams for the crown. As the equivalent value for a troy pound of gold was £46/14/6 (934.5 shillings), the implied ratio at that time was about 14.16:1. This significantly overvalued the silver.
But the UK was on the gold standard. From 1816, silver coins were only legal tender up to 40 shillings. The silver coinage didn't have to be worth its value in silver any more than modern coinage has to be worth its value in copper and nickel. It was the value of the gold that was important.
I think a key point when you compare the two is that the amount of silver didn't matter, because the exchange rate was principally set based on the gold standard. After 1816 in the UK, and 1873 in the US, the melt value of silver coins could be - and often was - significantly below the face value. It should thus not be surprising that coins from different countries in this era might not have silver content in proportion to their relative value. After all, a modern quarter is about the same size as a modern UK 10p coin, but is not worth the same.
Sources: Wikipedia (particularly Silver Standard, Coinage Act of 1873, Great Recoinage of 1816), and the Guide Book of United States Coins ("Red Book") by RS Yeoman (I have the 2004 edition).