I’m nothing more than an American that loves to consume all things British, but it seems like the Welsh are a culture within the larger British nation while Scotland always feels like it is a separate entity that co-exists along with Britain. What about their respective histories has led to this blending and also not blending?
I think the difference between culture and political reality has been an important factor in Cymru and Englands history. A long history of Marcher Lords after the Norman conquest their and gradual conquest of Welsh kingdoms which themselves virtually never encompassed all of Wales culminated in Edward 1st creating a principality with his son as prince. From 1472 to 1689 Wales was administered in Ludlow by the Council of Wales and the Marches, after this date administration was at Westminster. Legal union between Wales and England occurred between 1536 to 1543 as part of this a shire system was created in Wales similar to England.
One issue was a lack of a Welsh University, it had been hoped that. University would be created in Aberystwyth in the 1600s but this did not happen until the 1800s. Consequently unlike Scotland with ancient Universities the Welsh gentry studied at Oxford and Cambridge. With all political advancement in London much of the Welsh gentry were subsumed to some degree into the English system.
Scotland has a separate legal system and is not legally as the same country as England and Wales. The process of assimilation of Wales became brutal with Welsh children beaten at school for speaking Welsh during the 1800s. Many other formal attempts at suppression of Welsh language which is a basis of much of cultural expression were made. In some ways the question is how did the Welsh resist this so strongly!
Of course England has been assimilating Wales since the first Angles and Saxons settled and started the gradual conquest of the British, with parts of Wales in Scotland being conquered by Scots from Ireland (Galloway part of the Welsh kingdom of Strathclyde) so Wales is much smaller than it was, but it was not a kingdom, a cultural and language identity divided into several kingdoms.
Wales was not a kingdom and differences between former kingdoms and subsequent regions are strong, for an American understanding that South and Mid and North Wales are very different places in terms of culture might given the small scale of geography be hard to understand. English as a language has a long history in some parts of South Wales and some industrial areas had a large number of immigrants from elsewhere in U.K. and further afield who would not have been Welsh speakers. These parts of Wales are no less Welsh but they express their culture rather differently than Welsh speaking areas, in terms of Welsh rugby no division exists, a unified nation can be seen during internationals.
I would suggest politics and culture are different, Wales has had a long history of resistance to assimilation by England and is a culturally very different place and now with the Welsh Assembly it is to some degree developing a difference politically, but legally they are still joined in one system, unlike Scotland. The rediscovery by Welsh academics of Welsh literature and language especially at Aberystwyth University, The development of the eisteddfod tradition. Lloyd George and the breaking up of landowning estates creating Welsh freehold farmers, the growth of non conformist churches, many strands to this story. Owen Glendower represents a strong expression of Welsh identity and self expression, I would not myself feel that Scotland has any greater sense of national identity than Wales. As to the politics of independence the question of economics is a factor. In fact Scotland voted against independence, but the number of voters in Scotland who believe independence is viable are greater than in Wales. This may perhaps represent a sense of economic reality rather than cultural values.
I have to firmly disagree with your premise here. Both cultures have degrees of separation from "British" culture, both are very definitely British cultures, and both part of the British identity.
Large proportions of both would identify as Scottish or Welsh ahead of being British, but most also identify as being British- there are very few people who do not. With regards to language, Welsh is far more commonly spoken than Scots Gaelic, it being a first language in many parts of Wales. Language being an important indicator of culture and identity, I feel this points to the opposite conclusion you have made.
As with all other areas of Britain, each has their own culture - indeed each has several different ones - and each shares it in part with the rest of the UK, and also shares culture from the rest of the UK.
I did decide to use the anglicised version, perhaps a poor choice but it seemed as if it might be more accessible as it was used often in many popular histories, I hesitated and immediately regretted it on posting!