As someone else put it, it's for making sure your wife doesn't get suspicious of the weird ads you're getting, and when she checks the browser history it's clean.
Meanwhile Google, your ISP, and the NSA all know you're looking at freaky old lady bondage porn.
Except some sites seem to use your IP, so if you're both using the same WiFi, you're going to get ads for other party. And for anyone else who used the same WiFi, too
It's handy when you need to make sure that someone else can access a url ok without having to sign in to the website or anything. If you can immediately see the page in incognito mode without signing in, they'll have no problem
I remember having to use an incognito browser for testing at work one time, and it felt very wrong to pull it up on my work laptop instead of the personal laptop.
That's a good trouble-shooting step, but it's not a solution. That's some bullshit, sorry that happened. Maybe try clearing your browser cache and cookies if you haven't already? Basically my reasoning is if it works in incognito mode and only in that mode, then there's probably some saved state that the website is getting snagged on (state that a new incognito window wouldn't have).