Being a citizen of a Schengen country provides no benefits, the Schengen rules mostly impact people from elsewhere in the world. Being an EU citizen on the other hand has many consequences. Using that search phrase should give you much more information. Two big things come to mind:
- Much simpler formalities and stronger right to stay in the country. Basically, if you have a job, you can stay (without conditions like a minimum salary, education, language abilities). And if you lose your job or become disable later on, you also enjoy stronger protections.
- Easier rules to have your non-EU spouse and children live with you. That one is especially relevant in Denmark, which is one of the most restrictive countries I know in this respect.
And those rules apply to all EU countries, not only Denmark, so you can come and go without going through the whole visa/residence permit process or worrying about being able to come back to Denmark later on (that's the whole idea behind the “freedom of movement” and the reason why it's much more than a lenient immigration process).