apt-cache search doesn't use a fancy full text search engine like Synaptic does, but simply prints all packages where your search words occur in the package name or description.

As neither the package name nor the description of software-center contains the word ubuntu-software-center a search using apt-cache will never find it that way.

However you can use multiple search words with apt-cache search and it will show you only the packages that contain all of them. So you can use e.g.

apt-cache search ubuntu software center

which gets you a short list a packages including software-center.

BTW, the apt-cache actually interprets the search words as regular expressions.