Really, read whichever first. The situations and characters are all so different it doesn't matter. The Hunt for Red October was my first book by Clancy. It does mention 'the incident in London where Ryan stopped IRA terrorists from assassinating the Prince of Wales', but it doesn't go into so much detail that it ruins reading Patriot Games. (In fact, readers had to wait years in between Red October and Patriot Games to get details on how Jack got himself knighted by the Queen.)
It's hard for me to choose which book is my favorite. The Hunt for Red October has such an interesting concept (defecting Russian officers and the 'catapiller' sub system), but Patriot Games took a look at pre-September 11th terrorism and it's effects on the United States, as well as the politics of the IRA compared to the Soviet Union. Perhaps, since I'm more interesting in Ireland than in Russia, I find the factual portion of the novel more interesting, but I have reread both books several times.
"You're either lying or stupid." "I'm stupid! I'm stupid!" Megatron and Starscream