Hi UnixRonan. I had your same question myself a couple years ago as I started experimenting with higher end brewers, cold brew, entry level espresso machines (Brevilles are a standout) and finally made it onto the pour over "third wave" bandwagon.
caiobrighenti is absolutely correct about pour over giving one more control over variables like brew times, giving freshly ground coffee beans time to "bloom" etc.
I thought pour over coffee would taste exactly like that from a manual brewer. I mean it is coffee and water, right?
But it truly does taste different.
My least favorite method of brewing coffee, other than a moka pot, is the french press. They make a heavy, oily cup of coffee with some sediment at the bottom of the brewed coffee. Some folks love that.
Others prefer a much more filtered coffee that is not as dense and heavy as french press; more akin to what you are making in your drip machine.
Then there are those of us that fall somewhere in between those two camps; or those who like to experiment or who want the option to brew both a heavier cup and sometimes a lighter, cleaner cup.
That is where the pour over comes in.
I will admit that a decent burr grinder and the method of brewing coffee most definitely affects the flavor profiles and the quality of a cup of coffee.
But to be honest, though I have never tried a Chemex, I have used several other pour over brewers and I can not tell the difference. At all.
Gooseneck kettle. Experiment with grind size and water to coffee ratio. Paper filters. Those all made a difference, but after all that was said and done and dialed in, all pour overs I made tasted equally fantastic.
I ended up returning a couple of the more expensive pour overs (that looked real nice) and ended up with, and to this day happily still use, a 4 dollar plastic melitta.
I ended up giving away my Bunn coffee brewer a while back (which I LOVED) because pour over gave me more control and what I perceive to be a cup of coffee that equaled or bettered what I got from my old Bunn brewer (which I loved, remember).
Have not once regretted not having my Bunn.
My point is that since you are asking the question, you are at least acadmecially, as you said, curious about it.
I propose you continue using your coffee maker for your tea, but go ahead and buy a melitta. Shouldn't take too long to save up for one...in fact, go ahead and pay for it out of the change you find underneath your couch cushions.
If you already have a halfway decent grinder (no blade grinders...ever!) and do not mind shelling out a few bucks for a goose neck kettle, you can join our exclusive club of bandwagon jumpers that openly point and laugh at people at starbucks and pity "pour" souls like yourself who leave it to a plastic or metal monstrosity which spits and gurgles out water from old hoses/pipes that may or may not be clean and free of old gunky coffee sediments. If you happen to live out west, do not even get me started in on the calcium deposits that are mucking things up inside the hidden workings of your contraption.
You know how you like your coffee better than some hipster barista or some electric appliance does.
You get a pour over, and you know who makes your "perfect" cup of coffee? You do!