What was the military strength of the participants in the Falklands conflict?

The data on the forces involved are readily available on the web. For example, for composition of British forces you can look here and here. For Argentinian forces question has little meaning because not all available Argentinian forces were involved. For example, after the sinking of Admiral Belgrano the Argentinians decided not to involve the rest of their navy. At different times different forces were involved, of those available in principle.

1. British army was much better equipped and much better trained.

2. The British used a nuclear submarine. Unlike an ordinary submarine, this kind of submarine can stay submerged for long time, and thus there is no way to detect it by a radar. (This is an example of huge British technological advantage, and after they employed it, it become clear that Argentinian navy is useless in this conflict.) Radar was mentioned in the first version of your question. Then you edited and asked about sonar. This also does not help much against a modern nuclear submarine. It produces very little noise, and even if you detect it, there is almost nothing that a 1938 cruiser can do against it or to protect itself. So the Argentinians had to rely on their aviation only.

Americans helped the British with satellite reconnaissance, another technology not available to Argentinians.

Still, there were problems (for the British). Their main disadvantage was that the theater was very remote from the British bases while relatively close to the Argentinian (continental) bases. So the Argentinians could use land based airplanes and missiles, while the British had to rely only on their carriers and ships. One British attack was performed by their ancient long distance strategic bombers (even for them, this was an extraordinary feat at such a distance. This was the longest-range bombing mission in history at that time).

Question:
I find it interesting that 1980's UK had a more capable army than Argentina, the UK being a peacetime European country (albeit with considerable offshore possessions...) compared to Argentina with a ruling military Junta and various ongoing local conflicts with neighboring countries.

The UK traditionally is among the top 5 countries with their commitment to defense as measured by military budgets. This is both due to their historic defense culture and the threats that culture is safeguarding the country from. In 1982 the Cold War was still in effect and the UK was and remains an important NATO ally, then against the Soviet Union.

More importantly though is the UK's ability to project power removed from their borders as demonstrated by the Falklands campaign; a capability which the UK still maintains. The UK remains one of the top 3 countries in the world with this capability behind only the US and France. To project power 1000's of miles removed from your borders is really a capably not many countries possess. China for most of their post WWII history while having one of the largest army's on earth at times, has never possessed the military capacity to project force just 100 miles off their coast and successfully engage Taiwan.

Argentina traditionally is not among the top 20 countries in defense spending. Still in 1982 Argentina had about 220 first and second line combat aircraft to call on and a primary concern of the UK was they only had 40 navy harrier aircraft in their fleet sent to the Falklands. Worse because the fleet operations went 24 hours only 20 Harriers were available for operations during the peak of the conflict in April and May of 1982 at any given time.

The Harriers were new, untested in military action and subsonic. That was a major concern in that conflict for the UK. Turned out unfounded though with the UK knocking out about half the Arginine air capabilities within days of the UK landing May 21. Their Harriers performed very impressively.

I remember the Falkland's war. The Argentine military used French Exocet missiles to great effect. The British to their credit proved the effectiveness of their subsonic Harrier jump fighters even against Argentina's faster Mirage fighters.

Britain's primary difficulty was they had no fleet carriers only smaller jump carriers. This meant the UK could not use their AWAC's (Airborne Warning and Control System) planes, which can view hundreds of miles of airspace surrounding their fleet, to safeguard their ships from Argintina's Air Force. The AWAC's unable to launch from the smaller carriers. The UK had to rely on less capable radar which left their ships vulnerable to Argintina's fighters pop'ing up on the horizon and shooting off anti ship Exocet's at them. This proved to be a costly compromise, as the UK lost 8 ships sunk, another 7 severely damaged and had to be retired. Once the UK took the Falkland's back however they brought in their more sophisticated AWAC planes and based them on the Falklands airbases and Argentina never again successfully threatened.

One other interesting bit about that war was that the United States declared themselves neutral and sent the American Secretary of State, Alexander Haig off to conduct shuttle diplomacy to avoid a conflict. In the background though the US told Thatcher she had a blank check for any aid she needed. The US refuelled the British fleet on the way to the Falklands, provided satellite information on the Arginine military. The provided then sophisticated stinger missiles which allowed individuals to shoot down Argentina combat aircraft, and finally they provided the munitions which the Harriers would use to engage in the coming air war. Advanced sidewinder missiles, all delivered to the UK's fleet while in route to the conflict.

CIA files reveal how US helped Britain retake the Falklands
President Reagan at first said the US would be impartial in the conflict between two of its allies. But on April 2, 1982, the day of the Argentinian invasion, he sent Mrs Thatcher a note: “I want you to know that we have valued your cooperation on the challenge we both face in many different parts of the world. We will do what we can to assist you. Sincerely, Ron.”

Was the British army actually larger/better equipped/better trained?

Yes, definitely. The British had a professional army, while the Argentinians had a (mainly) conscript army. The actual size of the armies involved is not relevant. Both had obligations elsewhere, only parts of their respective armies were involved in the fighting.

But there is more. Argentina and Chile aren't the closest of friends, to put it mildly. There was a real chance Chile would side with the UK. Therefor the Argentinians placed their best units close to the border with Chile, not on the Falkland islands. That would be in the area of Ushuaia which is disputed by Chile and Argentina. Their best and cold weather trained troops were here.

Next, they used a lot of conscript troops from the warmer areas of Argentina to do duty on the polar islands in the south. Not a good idea. Have to say, the conscripts did much better than everyone expected.

To give you an idea how this affects the troops: I live in Thailand, where +25 C is considered a cold winter. People die when the temperature drops under +15 C. Not because they literally freeze to death, but due to exposure and unfamiliarity with this kind of severe cold. Now, send conscripts from such a climate to do duty in a place where +15 C is considered an unusual heath wave. Those poor boys must think they have been assigned a spot in hell (the Nordic variation).

I read that a British sub was shadowing the Belgrano for two days before striking. Were the Argentinians not using sonar to detect subs they knew the enemy has?

They probably did. But what kind of sonar? The General Belgrano was commissioned in 1929 as the USS Phoenix. It's not impossible to have the latest and most up to date sonar equipment on board such an old ship, but not very likely either.

The HMS Conqueror is a nuclear powered submarine. Maybe not the latest or the newest (launched 1969) but nevertheless the Argentinians would need much better sonar equipment to track her.

I find it interesting that 1980's UK had a more capable army than Argentina, the UK being a peacetime European country (albeit with considerable offshore possessions...) compared to Argentina with a ruling military Junta and various ongoing

No, more the other way around. The UK has been involved in nearly all major conflicts of the world since 1900. Argentina in none of them. The UK had the expertise and experience, Argentina hadn't. The UK was (and still is) a dominant world power, while Argentina never was. Not even the dominant regional power.