Synaptic is a pretty cool package manager that let you install not just the same metapackages other software installers allow, but also individual packages that may not be part of any other metapackage; Think of it as a GUI for apt-get. Starting some time when Unity was merely a concept, Ubuntu quit providing this invaluable tool by default, and Ubuntu MATE is no exception.
Here's what you need to do in order to get Synaptic working like you can in some other variants of Ubuntu and Linux Mint:
sudo apt-get install synaptic apt-xapian-index sudo update-apt-xapian-index -vf
The last command is in case post-install doesn't automatically handle this; if it does, then this command will go to the index update in progress instead.
The second package shown will install the Xapian index that Synaptic relies on, essentially restoring the "Quick filter" functionality that is missing if you installed Synaptic by itself.