IMHO, as someone who currently works as a photographer and who's degree is in video and film production...
Benefits of shooting AND editing in 4K:
- More "future-proof" if you plan to keep the footage you get
- Better resolution, in general should give a sharper and more detailed image*
- Ability to pull 8 megapixel images from video*
Benefits of shooting in 4k but editing in 1080:
- Ability to crop into the image up to 2x without noticeable pixelation (jagged blocky edges), so you could zoom in on a point of interest, and even make it appear the camera could pan or tilt a bit
- Should give a sharper and more detailed image*
- Raw footage is "future proof", hard drives have come way down in price... save it all!
- Finished edits take up less drive space, upload easier and faster
- Not that many people have 4K computer monitors yet, or 4K TV's (if they even watch videos you post on their TV to begin with- most YouTube etc is viewed on a computer, tablet, phone... delivering 4K to these devices is currently WAY overkill!)
- If you have the software and know-how, the ability to crop into the 4K image also means you can do some image stabilization in post
- Ability to pull 8MP images from (original) video
Drawbacks to 4K:
- File sizes are larger, need bigger memory cards, more hard drive storage
- Requires more processing power when editing (or at the very least more patience!)
- Most videos are watched on monitors, TVs, tablets, phones that do not have 4k displays
- Lens quality becomes more important with a higher resolution image
- Focus becomes more important with a higher resolution image (the clearer the in-focus details are, the more obvious it becomes when something is out-of-focus)
- Generally speaking, small camera sensors that capture higher resolution aren't as sensitive- and don't perform as well in low light situations
- Compression and BITRATE!!!*
* Important factor to consider = Compression and Bitrate
Compression is, in simple terms, the method used to take a large amount of data and squish it smaller. There are a few different resolutions of "4K" but the most common is 3840 × 2160, also called Ultra-High-Definition or UHD. If you take the pixel dimensions and multiply them you get megapixels... 3840 × 2160 = 8,294,400, or about 8 megapixels.
Look at the file size of any 8MP photo camera, and you'll realize the dilema with video... video is at least 24 (in the US, 25 elsewhere) frames-per-second. An 8MP image would be 24 MB uncompressed, but an old 8MP photo camera might actually produce around a 4MB JPEG file (JPEG is a type of image compression). If your video camera captured that same quality, you'd have 4MB x 24 fps or 96 megabytes-per-second of video?! Plus audio
Obviously that would be madness... so the images need to get compressed. Most methods of compression intelligently throw away data... "intelligently" meaning they are supposed to take away data in as imperceptible way as possible. Some do this better than others... and some throw away more than others.
BITRATE! - This can be a more informative measure of quality for a video camera than resolution or what codec (compression method) is used. This is basically the size of the file being created... the bigger the file, the more room for information there is for each frame of your video (and audio). There is lots of info on the web about all different cameras and I'm sure you can find the bitrate(s) for any camera you're considering. The bigger the bitrate, the more information the file CAN hold (provided the camera's sensor or lens or processesor is capable of producing that much information... otherwise you get a big fat file with no more value in it)
Why it matters? Well if a 1080 camera and a 4K camera both capture video at a bitrate of 60 mbps (that's megaBITS, not megaBYTES, per second)... and both cameras use the same form of compression to squeeze the image data down to that size... the 4k camera is squeezing twice as many pixels into that 60 mbps... which means each frame of the 4k image is being compressed MORE than the 1080 image is. Does that mean the 1080 image is actually higher quality? Not necessarily, but sometimes yes that can actually be the case!
Generally speaking, 4K cameras will be better than 1080 cameras... because they are newer. Not actually because they are 4k... but because 4k is a newer technology, so the image sensor in the camera, and the processor, and probably the compression codec, are likely better than what you might find in a similar 1080 cam.
Lots of generalities above... every rule has it's exceptions, so I'm trying to be accurate and specific but also not give bad advice. If I didn't help understand 4K, at least I cured your insomnia...
Shut up and fly.