Personally, I consider these to be two different, both equally valid, activities. And while I say and mean "different", they are not necessarily mutually exclusive to my mind.
To "take" a picture, as I think of it, is to capture something extant. Whether that's a facial expression, a pattern of movement (of anything, from flowing water to animals to various human creations)... Any moment (short or long) in time, that exists separate from the photographer.
To "make" a picture, as I think of it, is to set up conditions under which an image can be captured that meets a particular vision (or pre-vision, if you will) of the photographer. This could include everything from the simple choice of camera position, angle, focal length, focus distance, and exposure settings all the way through the elaborate creation of a scene, set, lighting, and what have you.
Often, a good photograph will have (in my opinion) been both. For example, an elaborate portrait setup - with a set (or at least a backdrop), lighting, costuming, hair and makeup styling, and the like, is a portrait that is made... But also, if the subject is given any control whatever over what they do within that setup, it is a portrait that is taken.
Other times, it may be closer to just one or the other, though I suspect it's almost always at least a little of each. A "made" still-life still "takes" from the objects arranged in it, and even a quickly "taken" snapshot has had choices "made" by the photographer, even if only where to aim and when to push the button.
There will be, of course, numerous opinions, often contradictory, on a question such as this. Having heard a number of them over the years, though, and giving it thought of my own, I hope that the above is an accurate reflection of the philosophy I have adopted with respect to this question.
If this needs any clarification, please ask in the comments, and I'll do my best to update it to increase clarity.
Thanks for asking an interesting question!