Two Software Center Icons 16.04

Remove the old version after upgrade

When you upgrade from previous Ubuntu versions, the "old" Software-center is not removed it seems. Since it is replaced by the new Software-center, you can remove it by running the command:

sudo apt-get remove software-center

This will not remove the new software-center, since it has a different package name.

If the icon still is there:

Most likely

Software Center was update to the latest version for 16.04, but the old .desktop file was left behind.

What to do if this is the case

• Browse in nautilus to /usr/share/applications

• Drag either one of the .desktop files with "software" in its name on to the launcher and try to start it from there. If it launches succesfully, that's obviously the one to keep.
  

  Note: the file name of the new .desktop file is org.gnome.Software.desktop, showing "Software" in nautilus, while the old one was named ubuntu-software-center.desktop, showing "Ubuntu Software Center" in nautilus.

• Test the other one to test similarly until you find the outdated one

Another option

is that for some reason, in the past you (or software you used) made a local copy of the global .desktop file. If the solution below gives no result, look in ~/.local/share/applications for a file ubuntu-software-center.desktop and remove it, log out and back in.