Depends entirely on the club. For Utd, we never want to even be playing in the conference league. Winning it would be embarrassing. Rival fans would probably use footage of us celebrating it as a stick to beat us with.
However, for a club like West Ham, it is quite an achievement, and gets them some recognition around Europe. So fair play to them.
League cup is a small win for us, but it couldn't ever be construed as anything but.
Again, for a team like West Ham, a league cup would be massive.
So TL,DR:
For us, League Cup - easily.
For a mid table team, either would be a fantastic accomplishment