What is the relationship (if any) between natto and miso paste?

They are completely different. They use different fermentation conditions and different cultures. The result is also totally different in taste and texture, with natto being slimy beans and miso being a paste. They are also used differently, with miso being more of a seasoning.

You can think of it as similar to two kinds of cheese, maybe emmentaler and camembert: they are both made from fermented milk, but the process is not the same, and the result is not the same either.

Natto is fermented whole soybean product, fermented with Bacillus Natto. To my palate, Natto has a complex flavour. Mostly it's the upfront flavour of the beans. But there is also a strong blue-cheese like note from the ferment. I really enjoy this part of it. Of course it would be remiss of me not to point out the slimy texture, which I like, but seems to be polarising.

It's possible to buy frozen natto in single-serving containers. If you eat the natto while it's still very cold, the cold numbs your taste, and a lot of the flavour can be missed. Please let it warm up a bit first. Also stirring it up will enhance the mucilaginous nature of the dish.

Miso Paste is a fermented, mashed combination of Soybeans and Koji Rice. Where Koji is a fermented rice using Aspergillus Oryzae. Different lengths of fermentation time create the white/red colourings. It's also possible (and common) to make Miso using other/extra ingredients, like Barley.

Miso is more often used as a further ingredient (or condiment), whereas Natto is eaten as a foodstuff in its own right. Both are fermented, and use cooked soybeans as an ingredient, but that is really the only commonality.

Miso is quite salty, with a subtle nutty soy-like flavour. A typical miso has around 20% salt by weight. The older the miso gets the stronger the flavour is. So a "white" miso has a lighter flavour than the darker "red" ones. I have a 18-month old batch that has been continuously fermenting (sealed in a vacuum bag). It's approximately chocolate brown, and it has a achieved delicious flavour. My friend's mum has been using the same big batch of miso for years.

One of my favourite uses is 50% miso & mayo brushed on top of grilled fish (or eggplants) before cooking.

Kingsley's answer is correct, however, there is also such a thing as "Natto Miso", sometimes called "Natto Miso Chutney".

This is usually a mix of barley, soy beans, a sweetener such as tapioca/potato syrup, and a flavouring such as ginger. All fermented with koji, as is plain Natto.

It's used as a garnish or condiment.