The Welsh have their own, distinct identity, along with their own, very distinct language. The Welsh language is still spoken, there are tv channels in Welsh, and the road-signs in Wales are written in both Welsh and English.
Curiously, the Welsh were one of the earliest, and you could say indigenous, inhabitants of the island now known as Great Britain. They would inhabit the territorries to the west of this island, but were also closely related to the tribes which inhabited what are now Scotland and Ireland.
In the Welsh language, England is often called Saxon-land. You may have heard the term Anglo-Saxon, which comes from the names of two different tribes, Angles and Saxons, which lived on the territory now known as "England", but were actually "immigrants". Besides the Angles and the Saxons, there were several other tribes too, all of whom had arrived from continental Europe, from what is now Germany, Denmark, France etc. Those "immigrant" tribes lived in the territorries closer to Europe - the south and the east part of Great Britain.
This means that the Welsh were pretty much a homogenous bunch, a group of similar people, while the English were actually a mixed lot of many and various tribes which originated from various places.
The Angles and the Saxons and those several other tribes had their own territorries at first, their own leaders, chieftans or kings. And they fought between each other, each wanting to assimilate the other tribe and its territory. Until one day there came a king called Alfred, who managed to unite quite a few of those tribes and amalgamate their territories somewhere in the 9th century. Alfred also fought further invasions from other European tribes, such as the Vikings, which were quite a nuisance with their raiding and pillaging. Thanks to Alfred, the kingdom of England began to take shape.
In the meantime, the Welsh also had their own little kingdoms, which occasionally had their disagreements between each other, but in the end they remained friends as they had a common enemy to fight - the Anglo-Saxons (or, if you wish, you can call them English). But still, the Welsh kept their distinct culture and language, although there never was a single, united "kingdom of Wales".
Which is to say, Wales was never "a country". It was a "region", but not "a region of England".
All was good for a couple of centuries, the Welsh and the English each keeping to their little corner of the island. Until the Normans (more pesky Europeans) landed on the island and took over "England". The Normans actually did a pretty good job in continuing the unification of all those little tribes in the east and south parts of the island, and created a well organised country, namely kingdom of England. Being neat and tidy, England also became quite powerful. And eventually, the English managed to conquer Wales, somewhere in the 13th century, and the English royals acquired the title "Prince of Wales". Today this is the most important nobility title in the country, after the monarch, as it belongs to the monarch's heir. For a couple of centuries the Welsh rebelled occasionally, but after the 16th century onwards, all went quiet and both sides played nice.
Technically, Wales joined England with an Act of Union dating from the middle of the 16th century. With it the English laws and English administration became valid in Wales and that was that. It's notable that the English language became the only language of the law courts and people who spoke only Welsh could not take public office. Since then, the more English one become, the more they could prosper, but nothing stopped the Welsh from preserving their language and traditions in their daily life. And they did that pretty well. Today around 20% of the Welsh can speak Welsh, and 57% consider themselves Welsh (and not English) when asked about national identity.