I have nothing but good things to say about Next.js, one of the few web libraries that I'm really excited about. Great developer experience, the API is simple and very stable and the feature set is really powerful.
Our experience was that nextjs was way more reliable (had to restart the nuxtjs local server every couple page loads), had much much better support for typescript, and had a more active community and ecosystem which led to questions being answered more easily and things improving at a quicker rate as well.
That being said, I think it's great to have competition and hope Nuxtjs continues to do their thing! I'm particularly excited about the new frameworks being built on top of or adjacent to nextjs (Blitzjs, Redwood, Remix).
I haven't used Nuxt, but I've used both Next and Gatsby a bit.
The biggest difference is that Gatsby is exclusively static, whereas Next can do static pages, server-rendered pages, or a hybrid of the two approaches.
Gatsby has a GraphQL-based data-fetching system, while Next is much less opinionated. Next gives you a place to do your data fetching, but doesn't dictate how you fetch it. Personally I find Gatsby's GraphQL stuff confusing, but YMMV.
I've been using Nuxt for about 18 months and on the whole it is fine, but I've had constant issues with pages trying to load out of date build files. Haven't ever experienced this with Next.
I've also found some of the libraries to be fairly flakey - PWA support breaks my app every-time I've tried to implement it. Lazy Hydration works for a while and then (without any changes) starts dropping random JS errors.
I've noticed a lot of valid Github issues being closed automatically with no attempt by the team to address them - they're simply opened up as questions on a forum that never gets looked at.