what effect does wifi really have onboard a plane?

This has been answered many times before so here goes one more time.

Arrivals and departures are when if somethings goes wrong it goes wrong fast and the crew of the aircraft want your FULL attention not playing some game or listening to your tunes.

Aircraft radios are amplitude modulated (AM) and are subject to a lot of interference, back in the early days (1960s) hand held radios gave off a lot of interference (they actually transmitted a small signal) which could interfere with the aircraft's receiver when the pilots were talking to a distant station. Close in to the airport the noise really shouldn't have been bad. Fast forward to the analogue cell phones and when the plane was down in the 5 to 10 K feet range the cell phones would start to handshake with their providers and this also caused noise in the pilots radio. Fast forward to today and the frequencies used by cell phones and the speed that modern processors run at there is very little interference to the aircraft radios. To see the interference you need specialised equipment and even then because the signal is digital you might miss it.

They also won't interfere with any navigational aid or cockpit instrument

Electronic devices do not actually interfere with the systems of modern aircraft. That is mentioned by the crew only to scare passengers into complying with regulations. The real reason that electronic gadgets must be turned off during certain phases of flight is simply that Federal air regulations require it, not because of any safety concerns. The safety argument is repeated so frequently that many people take it as gospel, but it has no basis in fact. No crash has ever been caused by the use of electronic gadgets (including cell phones) aboard an aircraft.

Gadgets typically don't even cause noise in a headset, unless they are very close by (within a meter or so) and are of a type to cause such interference. The classic example is a cell phone held within a few feet of a headset. But even pilots use cell phones in the cockpit without harming aircraft systems (even though they are legally required to turn the phones off just as passengers are). Phones in the cabin are too far away to cause noise in headsets as a general rule.

Anyway, the straight story is that regulations require that electronic gadgets be turned off unless their safety has been verified by the airline (which airlines never do). A FAA advisory circular allows airlines to fudge a bit on the law, which is why they allow gadgets to be used during cruise flight (the letter of the law actually says that everything must be turned off for the entire flight).

A FCC regulation also prohibits cell phone use specifically during the entire flight. This was motivated out of concern for overloading base stations on the ground, not out of any danger to the aircraft itself.

And no, gadgets are not prohibited so that you have your "full attention" on safety during takeoff and landing. That's another myth invented to rationalize the application of air regulations. But it's really just the law, not a technical or safety issue. Sometimes even flight attendants are taught these stories because it helps motivate them to follow the rules—a lot of people won't follow rules for their own sakes and require scary incentives to make them law-abiding.

It has no effect on aviation systems. Some airlines even allow in-flight wifi usage and sell you very overpriced wifi internet connection onboard. I hold a CPL pilot license, has been flying a lot, never turned off my cell or anything. The only case it can be disturbing is when the plane has just landed and all the pax start to switch on their cells and start calling. but once you are up and reached you travel altitude, no way it disturbs a single system...

A couple of mobile phones left on aren't going to cause problems, I've accidentally done it myself before, but if a lot people left their phones or wifi on it might cause a problem. Try using your mobile phone next to your laptop with a wifi signal sometime, my laptop doesn't function properly when I do that. It may or may not affect the navigational or communication signals, but it's better to be safe than sorry, they know a lot more about it than I do and turning my phone off is not a problem for me.

You think that annoying the pilot by causing interference on his headset is no big deal? How about when the pilot is asking about the best place to crash land in the next 30 seconds?

As I understand, often there is no problem. But, there's concern that wireless electronics could cause trouble at an inopportune time.

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It can make electronic meters and gauges move erratically and it can interfere with the microphone system. These things will not make the plane crash in a ball of fire :D

It can affect the navigation systems and other electronic devices that are affected by radio signals.