For those who don't know much about SK, Yongin is a city of over a million that's right next to/south-east of Seoul -- considered sort of the major suburb region. It's about as large as Seoul area-wise, but is one of the fastest growing urban areas in the country. Thus it has generally better infrastructure planning, roads, newer buildings, slightly more affordable housing, etc. making it very desirable as a location for families to live in with not terrible commutes into Seoul if needed (about an hour).
Seoul, while an amazing and dynamic city, can be a bit of an urban snarl. The layers of urban development, especially the post-war decades can be a bit dodgy, expensive and not all that great to raise a family in.
Living-in, or commuting to Yongin is far more desirable than Seoul at the moment.
Comparing "Seoul" to "Yongin" is not very good in the grand scheme of things. Seoul has a metro area which doesn't include Yongin, but it would be unwise to consider Yongin outside of Seoul since it's basically a satellite city connected to Seoul. The name for this is Gyeongi but that's another thing. Most of Gyeongi is still basically Seoul in natural form. The same way Greater London is part of London but separate from the City of London.
Because Seoul is so big, it's more akin to calling Yongin a "suburb" of Seoul, and Yongin has its own suburbs.
Seoul and Yongin compared area wise would be the worst since you have to first set the definition of Seoul. Seoul covers Yongin in most natural definitions of a city, but in reality the Seoul inner region is its own and is obviously tiny. It's better to call it Seoul metro area (Seoul Capital Area).
Yongin is an old city, it doesn't have better infrastructure planning at all, but it doesn't have a lot of super old infrastructure which is great for future planning. Be it due to budget or just slow crawl of improvement, who knows. It's desirable because it has everything you need, now including Costco, IKEA, etc. and it's got land. That's all.
Your last answer is weird. There's nothing better about Yongin than living in Seoul, except for possibly commutes. It's all a matter of perspective and negatives. I don't think either are objectively better but living in Seoul is definitely more sought after and "higher" status.
Seoul, while an amazing and dynamic city, can be a bit of an urban snarl. The layers of urban development, especially the post-war decades can be a bit dodgy, expensive and not all that great to raise a family in.
Living-in, or commuting to Yongin is far more desirable than Seoul at the moment.