tl;dr- mechanical feels nicer but won't change your ranking one bit
mechanical vs membrane (not electronic)
Most of the difference is about the feel of pressing a key, mechanical will usually feel smoother, or have specific feel to it. (Like with brown and blue switches where there's a mechanic in it to make textile/audible click.)
But mostly no difference in play, you push a key, and something happens. With membrane you may not know exactly when you've technically 'pressed' the key. You might push it down, but not actually hit it all the way so it hits the switch under it. It's easier to tell with mechanical.
But as long as you're actually pressing the key to the activation point, mechanical will not help you play 'better'.
However, if you're working with an old (like 5+ year) membrane keyboard, some of the more commonly pressed keys (QWER...) may have started to wear out and not function. Mechanical ones will generally last much longer and still function perfectly, but again, talking 10,000's of of 10,000's of keypresses, so as long as you buy a new $7 keyboard every couple years this won't matter either.
Mechanical keyboards allow for more simultaneous key presses (I think membrane caps out at 12? so like, literally face-rolling levels, isn't going to matter in LoL, might in some mmo's...) So, again, not going to help you play 'better' in LoL.
But, mechanical keyboards do feel much better on the fingers, once you get used to the weight/pressure difference, and pick the right type/color of switches for your preference.
I have a mechanical and I would hate to go back to membrane just for this reason, they feel so icky to 'click' now. This matters far more for typing than playing LoL.
Downside to mechanical is cost, and they click louder (even the 'quieter' switches like red/black, they're still much louder than any membrane, so if you have roommates and play at night...consider that part carefully. Yes they are that loud that you could piss someone off using them.)
edit: for what I use-
https://steelseries.com/gaming-keyboards/6gv2
steelseries 6gv2 black switches
They are 'heaver' than most (the actual keyboard and pressing down the keys)
But I've had it for 4? or more years now and it shows no signs of wear at all, and I still like it. (If I cleaned it up really nice/dusted it out/etc I'd bet you couldn't even tell the difference between mine and a brand new one.)
But it's huge and heavy (for a keyboard)..so not a good portable one if you use a laptop/have keyboard on your lap when you play/travel to friends houses to play, etc) Also no backlighting or fancy stuff a lot of newer style gaming keyboards have. It's just a keyboard, good, reliable and plain.