Why was the Vulcan bomber used for the Falklands raid?
This is borderline as an aviation question but I will answer it anyway. 

They didn't use those weapons because they did not have them at the time:

Conventional cruise missiles did exist but most were nuclear, and the UK didn't possess any conventional ones. 

ICBMs with conventional warheads did not and probably still do not exist because they are not a good idea:


Their capacity is low: A sub launched ICBM has somewhere around 2.5 to 3 metric tons' throw weight, much of that would be taken up by warhead guidance, thermal protection and other necessities so your actual warhead would be maybe 1000kg, or 2200lb. That's not that big a conventional bomb, and it could only launch one. A fighter bomber could drop 2 of those. The Vulcan bombers on the raid carried 21 1000 pound bombs. 
They aren't accurate enough: a nuclear weapon can be a bit off and still destroy its target utterly, a conventional weapon needs a great deal of precision. The Polaris missile had an accuracy of 3000ft (900m). That's okay for a nuke, it' still ruin someone's day, but completely inadequate for a 2000lb bomb. More modern ICMBs have more like a 100 meter accuracy, which is a lot closer but still not good enough for a conventional bomb
They are expensive: that's a lot of cash for a 2000lb bomb that probably will miss
Launching one could start WW3: If you launch an ICBM someone is going to see it and if they think it might be directed at them a retaliatory strike could be the answer. The world has come perilously close to this more than once, it simply isn't worth the risk!