Carl does sort of recontextualizes the Guardian of Forever quite a bit.
In TOS, the Guardian appears to be in ruins, sending out ripples to draw the Enterprise in to investigate. It has been waiting for a question, and seems to be just an advanced computer willing to answer anything and send the crew anywhere regardless of the outcome. It doesn't seem capable of changing the order in which it presents events, although this is debatable based on information from TAS.
In Discovery, Carl can appear at will (although he is detectable by lifesign readings), and is on a desolate planet in the middle of nowhere. He speaks in puns, in jokes, and appears to be an old man reading a newspaper straight out of 1930s America for some reason. He has a portal through time and universes (or at least Georgiou's mind) that he can control at will, sending people into their own consciousness apparently (a la Q's illusion in All Good Things...).
They seem to be two wholly different creatures with wholly different powers, but the self-epithet "the Guardian of Forever" does imply a much more powerful creature than TCOTEOF presented us (and more respectable of the timeline than the Guardian's blasé attitude showed). With Carl, Terra Firma shows us a being who is acting much more like one would expect the Guardian of Forever to act.
I'm assuming that Terra Firma 2 will give us the actual Guardian reveal and explain away some of the perceived inconsistencies. Burnham and company don't know anything about the Guardian of Forever, but Vance and co. should know all about it from their historical databases when it was used for a couple years at least for time travel research.