There are few things which annoy the Welsh as much as the common conception that Wales is part of England, particularly when it comes from people who quite happily recognise the fact that Scotland and Northern Ireland are not. This is not wholly without reason, however, because the exact nature of the relationship between Wales and England was a rather vexed one from 1284 up until the second half of the 20th century.
Unlike Scotland, where the Union was approved by both countries' Parliaments, Wales was simply annexed by the Crown, first under the Statute of Rhuddlan of 1284, and then the Act for the Government of Wales of 1536 (usually referred to in popular usage as the Act of Union, though this is strictly speaking incorrect). Both of these refer to Wales being incorporated into England:
"The Divine Providence, which is unerring in its own government, among the gifts of its dispensation, wherewith it hath vouchsafed to distinguish us and our realm of England, hath now of its favour, wholly and entirely transferred under our proper dominion, the land of Wales, with its inhabitants, heretofore subject unto us, in feudal right, all obstacles whatsoever ceasing; and hath annexed and united the same unto the crown of the aforesaid realm, as a member of the same body." (Statute of Rhuddlan, 1284)
"...the dominion, principality, and country of Wales justly and righteously is and ever hath been incorporated, annexed, united, and subject to and under the imperial crown of this realm as a very member and joint of the same..." (Act for the Government of Wales, 1536)
For over five and a half centuries after the Statute of Rhuddlan, "England" included Wales both in law and in the general English imagination. Welsh law was abolished by the 1536 Act and replaced by English law. Welsh courts were abolished by an Act in 1542, and replaced by a system substantially similar to the English judicial system. The Acts of Union of 1707 speak of uniting the Kingdom of England and the Kingdom of Scotland, with no separate mention made of the Prinicipality of Wales. And up until very recently, every Act of Parliament applying to both England and Wales only used the term "England" (in stark contrast to statutes that also applied to Scotland, which always said "England and Scotland"). The most stark reminder of this is the oft-quoted entry in the index to the original edition of the Encyclopaedia Britannica: "For Wales, see 'England.'"
The Welsh never accepted being treated as a part of England, but in general, little notice was taken of this until the 20th century. The first recognition that that Wales was, somehow different from the "rest of England" did not come till 1920, when the Church in Wales was separated from the administrative control of the Church of England and made an independent province of the worldwide Anglican communion. It took even longer for this recognition to make its way into civil affairs, and it was only after Labour came to power in the post-1945 period that concessions to Welsh national identity began to be made. These were minor at first, involving things like the Wales Gas Board, created in 1947, and the Regional Hospitals Board in 1948. Constitutionally, the deciding moment only came in 1955 when Cardiff was recognised as the capital of Wales. This was as close to a formal statement of de-annexation as the United Kingdom was likely to get, and finally recognised Wales as a separate nation within the UK.
Since then, laws and government documents have increasingly used "England and Wales" where they would formerly only have used "England". For example, when statutes intend to confer the power to make rules to apply to both England and Wales, they now say that rules for "England and Wales" may be made by the Secretary of State with the concurrence of the Lord Chancellor. Before the 1950s, they would have just said "rules for England". Most cross-border institutions now have "England and Wales" in their title (England and Wales Cricket Board, the Law Commission of England and Wales, the General Register Office for England and Wales, etc.). And, of course, this position has been finally confirmed by devolution with the creation of the Welsh Assembly. Wales, finally, can say that it is not a part of England.