Spider-Man is a real menace. If you see him across to table you should know exactly what he is capable of. If you are taking him yourself, hopefully this will help you get the most out of him.
Know thy self

I have to admit, when I was first learning the game I was rather underwhelmed by Spider-Man. His attacks aren’t as powerful as a lot of the other 4 threat options and he doesn’t have as obvious defensive power as, say, Captain America. But I’ve learned that Spider-Man – used right – wins games by himself.
If you throw him at the enemy getting involved in a big melee scrum, you will probably be consistently underwhelmed. There are other characters for that.
Where Spider-Man excels is in the mid to late game. Get a few power on him and he will be using Web line and web-a-pult late in the round to push and pull other characters off objectives, daze Extract objective holders and swoop in to pick up the objective they were holding. It’s not hard to envision a situation where a six – or more – point VP swing happens in his one activation.
He is particularly effective activating late in the round, where you can pull characters off objectives without them being able to move back.
With great power…

All of this leads me to the most important point. Spider-Man with a big stack of power has a huge amount of options and will really shine. Spider-Man without much power feels over-costed and ineffective. So the key is making sure he has the power he needs.
Opportunistic strikes in turns 1-2 are the best way to do it, and then possibly taking some damage in return. But only having 5 health can make this risky. Don’t let a proper bruiser get two attacks on you as you probably won’t survive that.
The trouble is it is so tempting to use him as a wide objective hound. Turn one is often him moving up and using his one power to grab an extract objective. Often this is such a strong line of play that it’s the right choice, even though this can make him feel “useless” for a few turns. But this is often a necessary price to pay.
The original Wallcrawler

That leads me on to highlight another reason to love Spider-Man. He is currently the most mobile character in the game. A few characters have L move, but none of them ignore terrain. Black Panther comes close with his pounce ability giving him a greater range in theory, but one you have a decent amount of size 3+ terrain in most real game situations, Peter is faster.
Team ups
Because power is so important to him, affiliations that give power or reduce costs are what he really wants. So Wakanda doesn’t really help him out and you will already be including Black Panther – see below.
Of Cabal and Avengers, Avengers plays best with him. Cabal requires you to do damage with an attack to generate your power. Peter can throw punches, but it’s not why you included him on the team. A passive cost reduction feels really good. Web line for 1 power is great.
My Spidey-sense is tingling

I’ve written about this before, but it’s worth reiterating. Spider-Man has great defense. His Spider-Sense ability makes his threes across the board better than fours across the board. The only thing holding him back from being a real defensive power house is the rather average 5 health on each side.
Spider vs Panther
So now the biggest reason you might not include Spider-Man in your roster: Black Panther.

T’challa is almost as mobile, has push-pull shenanigans, great defenses, and hits a bit harder. A tough call then which to include. That is until you realise that Black Panther comes with the Avengers affiliation where Spider-Man doesn’t. That makes team building a lot easier. Cap, T’challa and Black Widow is a great versatile base for an Avengers team. It allows (typically) 7 points of flex to include some heavy hitters, but it has a really solid base for scenario play. If I swap T’challa for Spider-Man, he comes with an opportunity cost of having restricted options as you are forced to include another Avenger. You might be fine with that, but it adds more limits on squad building that you can avoid.
It’s a tough one, but which of these two characters – or possibly both – you chose to include will depend on which characters you like and can get the most out of.