Why Great Britain or England/Scotland/Wales/Ireland in different sport?

The accepted answer, while doing a good job of presenting the background from a British point of view, doesn't really get at the actual reason for the unusual status of British teams in some sports.

It's very simplistic to consider that the UK's “constituent countries” have “full country” status as far as international law or sports bodies are concerned. Many sport bodies (including the IOC or FIFA) have in fact made their rules in this respect considerably stricter in the last decades and would almost certainly refuse to acknowledge anything like that if the UK (or any other country) would request it now.

There is therefore no reason to assume that the UK could simply choose to be represented by separate teams at any time, e.g. in the Olympics. Beyond all their peculiarities and no matter how the locals want to see them, the UK constituent countries are in fact very much in the same position as Bosnia and Herzegovina's autonomous entities, Swiss cantons, Belgium communities/regions, Dutch or Danish overseas dependencies, etc.

The actual reason England, Wales, Scotland and (Northern) Ireland teams still exist in rugby (union or league, though there is also a Great Britain rubgy league team), football, curling and cricket is that Great Britain had such an overwhelming influence on the development of these five sports, that it historically enjoyed, and to some extent still enjoys, an outsize influence on their governance. From that point of view, what's special here is not so much the way the UK is organised rather than the history of the sport itself.

On top of that, the all-Irish team in rugby union represents another anomaly, also rooted in history. I don't know much about the details but this team existed prior to the independence of Ireland and rugby was at the time primarily a sport of the protestant middle class. The latter might explain why there is no team for Ireland like there is football and other sports.

The United Kingdom (UK) is not alone. There are more than 20 members of FIFA (the primary organization that runs international soccer) that are not independent countries as recognized by the United Nations. For example, Puerto Rico, American Samoa, and the U.S. Virgin Islands compete separately from the United States even though they are U.S. territories. Hong Kong and Macau have their own teams despite being special autonomous regions of China. In addition to the four you mentioned, the United Kingdom has several more teams that fall within its large historical umbrella: Anguilla, Bermuda, British Virgin Islands, Cayman Islands, Montserrat, Turks & Caicos Islands, and Gibraltar.

When I began looking into this question, my assumption was that FIFA liked maintaining a separate sense of what constitutes a "country" than the U.N. because it likes to imagine it is an international organization with power and influence on the level of the U.N. (And maybe this is true to some extent) Then I read Luke Bradshaw's story of Paul Watson's quest to bring the tiny island of Pohnpei into FIFA. I discovered that "As a non-governmental organization, FIFA is legally obliged to accept membership applications for states that want to join." FIFA is allowed to create application requirements and they leverage those to stall applications they don't like.

Of course, FIFA is not the only organization the runs international competitions, the International Olympic Committee (IOC) is another. Unlike in FIFA, the UK sends a single team to the Olympics, not four constituent teams. However, they (almost) never send any soccer teams at all. That's because they are worried that if they played soccer as the UK in the Olympics, FIFA might rescind their ability to play soccer as individual nations in the World Cup. One exception to this was during the 2012 Olympics which were in London -- the UK took the calculated risk of playing soccer as the host and it worked out just fine.

There's a clear downside to playing separately. Two of the best soccer players in the world over the past decade have never played in the World Cup because of it. Kim Little and Gareth Bale have been superstars in club soccer but haven't ever been able to shine on the biggest stage of all because they were born in Scotland and Wales respectively and even their skills have not been enough to drag these smaller nations into the World Cup. That will change next year for Kim Little whose Scottish team just qualified last night for the 2019 World Cup in France with a 2-1 win over Albania!

The underlying "why" question remains. Why not play as the United Kingdom? The desire to play for Scotland or Wales or Northern Ireland or England instead of a consolidated United Kingdom must be rooted in national pride and history. It's maintained by negotiated agreement. The Home Nations agreement between the four regions sets out who is eligible to play for which of them and is much stricter than the general FIFA rules that allow quite liberally for (often Brazilians) to play for all sorts of national teams around the world.