Because Argentina knew that the UK wasn't going to nuke them over the Falklands.

The problem with nuclear weapons is that they are something of a bluff. Since World War II, no state has really been that eager to be the first to use a nuclear weapon in a conflict, even against a non-nuclear state where immediate retaliation (either by the non-nuclear state or a nuclear ally) is a likely option.

The general deterrence value of nuclear weapons is very unclear, even if it seems their ability to deter against other nuclear attacks is much better understood (though even that is less clear than most people realize).

The US, for example, implied during the first Gulf War that it might use nuclear weapons if Iraq set the Kuwait oil fields on fire. It wasn't clear that this actually influenced Saddam, however, who believed that the US was likely to use nuclear weapons anyway, but still ordered that the fields not be set ablaze. (They were set ablaze anyway by local commanders, without an order. The US obviously did not nuke Iraq in any case, though it did blame Saddam for the fires.)

It's worth noting that tensions were sufficiently high with the USSR as to deter any consideration of nuclear attack. The British navy did not even attack Argentine naval targets outside of the zone of exclusion.

The period of 1979 to 1983 is arguably the closest the world's ever come to nuclear war. A year later, Operation Able Archer nearly triggered a Soviet nuclear attack. In 1979, the Vela Incident and the Soviet invasion of Afghanistan had ratcheted tensions to high alert.

Galtieri's government made a miscalculation about the UK's commitment to a very minor territory. However, they understood (correctly) that Britain was on a knife's edge with the USSR.

As it was, Thatcher took considerable odium from the British Left over the sinking of the General Belgrano, an old WW2 cruiser. Information leaked to MP Tam Dalywell led to a bureaucratic inquisition of the MoD, and also, colorfully enough, several pacifist anti-war punk bands such as Crass.

Even if the UK didn't have nuclear weapons, it could still have inflicted massive damage on Argentina using conventional weapons (look at what the UK did to Dresden in Germany 40 years previously)

The UK never attacked Argentina beyond small scale commando saboteur operations. This was potentially due to a desire to contain the scope and scale of the war and to avoid antagonising allies such as Chile and the USA.

This article has some interesting information on plans by the UK to bomb mainland Argentina (using conventional weapons).