Had some feedback asking what the difference was and why I haven't mentioned any witches and witchcraft.
A Mage uses the enery interactions at the sub-molecular/quantum levels of physics to give them the power to do what is neccesary, their mind and imagination does the rest. The major problem for the Mage is controlling that power, get it wrong and the chances are they'll burn up from the inside out a bit like putting too much current or voltage through too small a conductor, assuming they are unlucky. Get it too wrong and they explode with devastating results. That's essentially why the apprentice/journeymanship process takes so long.
A Theurgist/Thaumaturge chooses/is chosen by a higher power (angel/demon etc) their power comes from their source and they are unnafected by the process. That means if they have a very powerful source they can do some spectacular things within and without the spiritual realm. Their host (for want of a better word) also protects them and can extend their lives. What a Theurgist/Thaumaturge has to do though is 'ask' for the power hence the chants which are requests for the hosts power to be used in a specific way. This limits them to an extent as it can take time for their full powers to be brought to bear.
Witches etc. are simply Theurgist/Thaumaturges who have a tie to a minor power. Their powers are limited and a Mage could deal with one fairly easily if they step out of line unlike a Theurgist/Thaumaturge who has ties to a higher power who is so protected by their host that a Mage could barely detect one, never mind raise the power to hurt one.
Hence Mages tend to stay well out of the way of the realms spiritual. Their are exceptions, but you'll have to wait for that in later chapters.