From my French point of view, I would like to add a few things to Mark's post.
First, the UK and France were rival countries for a long time (the UK is often referred in French as our “greatest enemy”).
After the WW2, the moral impact of the war made those countries to choose ways to protect themselves from another war. However, their responses were quite different.
- The UK chose to pair up with the real winner of the WW2, the USA. There are great ties between these countries (one of them being language) and from that time, the UK and the US have very similar exterior policies (the UK followed the US into the 2nd war in Irak and France did not) and they seem to form partnership (see the relations between Thatcher and Reagan).
- France thought the priority was reconstruction and security for the people. Thanks to the Marshall Plan's money, European countries were able to rebuild themselves and were pushed (by the US) to team up (though the US did not anticipate that the alliances between the European countries were to become a big union which is now an economical concurrent). Therefore, the two nations (France and West Germany) which were the most chocked by the war began to maintain a relationship with the idea that a strong relationship will prevent them from attacking each other forever.
Their paths were then very divergent. The UK was (and is still) perceived as being both in and out of the European Union at the same time and from the very beginning of this union.
For instance, the UK tried to create its own union to compete with the EEC (the organization which preceded the EU), the EFTA (note that the UK is ironically no longer part of this association).
When De Gaulle was president he fought till his death against the UK entering the EEC because he thought the US' influence on them would be damageable for the union and the future showed that he was quite right, the UK often views the EU as an economic union instead of an organization of countries seeking to make their voice count. This lead to some hard times between France and the UK, e.g. the well-known quotation from J. Chirac (at that time France's PM) about M. Thatcher: “What does that housewife want more? My balls on a tray?” (original: “Mais qu'est-ce qu'elle me veut de plus cette ménagère ? Mes couilles sur un plateau ?”). Thatcher had a very harsh policy towards the EU, she notably said “I want my money back”, and this sentence can explain a lot about the UK's vision of the EU.
Now, that said, to get back into the topic, France built itself inside of Europe and towards the goal of avoiding another European war at any cost. France, mostly because of De Gaulle's complex relationships with the US (Roosevelt did not trust him (fr) during the WW2 and the AMGOT did not ease their relations) did not trust the US in any domain.
De Gaulle made some decisions about NATO (because he thought that this organization aimed more at protecting the US than at protecting all the other countries inside of it) that pushed all foreign military forces to leave France's territory. Like Mark wrote, the UK and France were both inside of NATO during the Cold War but France was far from playing with only one side.
So, France built big companies (Thales, EADS) to create weapons by itself (and France is one of the biggest weapons exporter) and with the help of other countries of the European Union (Airbus and Eurocopter come from those partnerships) in order to keep its independence (and avoid some problems which could come from buying weapons from a foreign country, like espionage).
On the other hand, the UK maintain a relationship of trust with the US (Special Relationship) and rely on this country for their weapons. From the Cold War, France and the UK learned that they were not the World's ruling countries anymore (the Suez Crisis is a good example of this, as the US pushed the UK and France to give up on their ambitions). USSR and the US became what we call in French “les gendarmes du Monde” (the policemen of the World). Suez Crisis had a great impact, the UK decided to follow the rules imposed by the US most of the time when France chose to distance itself from the US and NATO.
To sum all this up:
- The UK knows that their power is not the one they had before the WW2 and chose to team up with the US in what is now known as the “Special Relationship”.
- Given US' dominance over the world and its policy, France chose to distance itself from the US and to rely on itself and on peers like the countries in the European Union (which, for some of them, may have the same state of mind) for its protection.
I tried to remain objective, I know that both the UK (WW1 & WW2, acquiring the A-bomb…) and the US (WW1 & WW2, the Marshall Plan…) helped France and the French people a lot, but this answer has to be put in the right context and some political decisions were made out of mistrust.
NB: Sorry for this long post which can be quite French-centered and contain mistakes.