After some experience playing both with and against this card, my conclusion is: it isn't as good as it looks, which is a bit concerning given that (at least to me) it only looked decently good rather than amazing.
The three main ways in which a piece of ICE can have a good effect are stopping power, taxing power, and the facecheck. But Mycoweb doesn't seem to do particularly well in any of those dimensions.
Let's look at the facecheck first. Jinteki generally only wants to spend 8 on rezzing one piece of ICE in the early game if it does enough damage to the Runner in the process to make up for the cost of the rez. Spending 8 is quite the tempo hit, potentially taking a couple of turns to recover from – but that's forgiveable if you land a hit on the Runner that also takes a couple of turns to recover from. In this case, though, an early-game rez of Mycoweb frequently does nothing at all: in the early game you normally need to install all the ICE you draw (meaning that it doesn't end up in Archives), if you spent 8 on a rez then you won't be able to rez anything else impactful even with the 2 discount, and you're unlikely to have another spiky sentry or code gate rezzed to copy a subroutine from. Some Runner decks opposite will keep the "period of time where rezzing Mycoweb is useless" around well into the mid-game; in one game I played as a Runner, the opponent rezzed a Mycoweb when I already had a decoder installed (hoping that I would pay the costs to break it), I judged that the subroutines would do less damage than the break cost and let them fire, and the Corp realised that none of the subroutines actually did anything useful in the gamestate at the time and had to let me past.
In the very late game, the facecheck would theoretically be more impactful, if you could get the subroutines to fire – maybe by that point in time you have a lot of highly-damaging subroutines to copy, and Archives might be loaded with ICE (especially if the opponent is trashing it). But at that stage of the game, the facecheck doesn't matter so much because the opponent will almost certainly have a way past the ICE anyway (especially because ICE installed late tends to be on the outside of a server, the most vulnerable location).
What about stopping power? If someone is trying to make the critical game-winning run through a Mycoweb, they can usually get through by matching 5 strength and breaking two subroutines (the last two). Mycoweb is in a weird spot where it has a lot of subroutines, but they rely a lot on synergy in order to work, e.g. the first two subroutines usually have no immediate impact other than setting up the last two subroutines (although they are helpful for future runs). In one of my games, I was trying to steal the last required agenda point from R&D with low credits and dubious breakers, and couldn't afford to break all the subroutines – so I just broke half the subroutines, and still got in. In the early game, the stopping power is even worse – you can't use a Mycoweb to stop a steal unless you have a run-ending sentry or code gate rezzed already, or rezzable from the play area or Archives, and you won't be able to afford both the ICE to copy and the Mycoweb. As such, the stopping power is somewhere between "somewhat porous" (late-game) and "this doesn't matter at all" (early-game). One way you can try to patch up this weakness is to play ICE like Anemone that make the first two subroutines relevant even on a last desperate run; but this is only a tax of 1–2 more, so the synergy probably isn't worth it unless your deck wants to play Anemone anyway.
I was initially expecting the taxing element to be the best part of Mycoweb, and it is, but it still isn't as good as I'd like. The basic issue is that the normal approach to taxing ICE is to put it on a server that the opponent's deck wants to run repeatedly (e.g. R&D against certain Shapers) in order to limit the number of runs that they can make there (via forcing them to spend more money, and thus more time repairing their economy between each run). In order to get value for that, you want to have your taxing ICE rezzed early – otherwise the opponent will get most of their value runs in the early game and just switch server once you've spent effort in fortifying the server they were originally attacking. But Mycoweb isn't useful for taxing until you're already set up with additional ICE to copy, or appropriate ICE in Archives, so it only taxes through a small proportion of the game – and that mostly negates the purpose of taxing the opponent, because the number of runs you're stopping is low in absolute terms and thus the amount of damage you're doing to their gameplan doesn't justify Mycoweb's 8 rez cost. The actual amount taxed is also smaller than it looks: decoders are generally more efficient at breaking things than fracters or killers, the opponent can know or guess that some of the subroutines might be irrelevant and not break it, and the opponent may consider face-tanking the subroutines in certain gamestates (often the best you can do with the last two subroutines is 3ish net damage, which most runners are capable of tanking if they don't have to do it too often).
I'm not yet sure whether Mycoweb's status is "playable as a 1-of in most decks which synergise with them, because it's decent in the late-game", "playable only in decks which synergise with it particularly well", or "never worth it regardless of deck" – I don't have enough experience with or against it to work out in which of those categories it falls. But I don't think it's a staple, and your deck would need to fit it particularly well to consider playing it at 3 copies.
I was excited for a criminal draw but it's hard to justify slots for this card. The draw effect is simply not worth it. You need to spend a card, 2 clicks and a credit to get 4 cards. A 2 clicks is worth more than 2 cards. One card replaces itself. So you can think of it as 1 credit for 1 card. The tag removal is very situational and the tempo gain is mediocre. This card would have any merit if it was 0 credits or was a run event with the options to choose after run is successful.
— ruby272