Different types of Nike Shoes
Beginning with the basics.
Nike products have a Swoosh on them
Adidas shoes have three stripes
Beyond the visuals, it is difficult to generalize. Perhaps 20-30 years ago you could have said that Nike was primarily a design company and Adidas primarily an engineering company, but that would not be accurate today. Of course I don't pretend to be neutral because of my long past association with Nike. But I would say that the Nike Sport Research Laboratory (NSRL), staffed by PhD's, does cutting edge research, much of it peer-reviewed, not only with respect to footwear, but also apparel and equipment. I sat on the NSRL's ethics oversight committee for my last couple of years at Nike and worked closely with them for many years before that, so I saw a lot of their work. It is very expensive to operate a facility like the NSRL and nobody in the industry other than Nike has anything like it. So the existence of the NSRL is a big advantage to Nike in creating state-of-the-art, top-of-the-line products that support peak performances by the world's best athletes.
The benefits of the NSRL work do find their way into the $80 shoes, but for casual athletes I would concede that the distinctions are less crucial.
In footwear, the most important attributes of a product are fit and comfort, and those are highly personal. The last of a shoe (the foot-shaped form around which it is built) strongly affects both fit and comfort, and consumers need to shop around for a shoe that fits them well. For 15-20 years before I went to work at Nike I wore only Nike athletic shoes, because their lasts and cushioning felt best for me. Some people prefer Adidas, Asics, Brooks, Saucony, whatever. I've heard it said years ago that Nike lasts run a little wider than Adidas lasts, but I don't know whether that is still true (if it ever was).
Nike does own a lot of patents covering the Nike Air cushioning technology, and other patents for other cushioning technologies including a favorite of mine, an extremely lightweight and very cushy foam called Lunarlon. Adidas is now out with their own new patented foam that they claim is much softer and bouncier than anything else on the market, but of course the softer a cushion is, the more work is required to walk or run on it, so it might be hard to strike exactly the right balance between the optimal levels of softness and energy return.
Stylistically, Adidas was out in front of Nike for awhile in collaborations with outside designers (for example their longtime association with Stella McCartney), while Nike still relies more on internal talent for fresh new designs. In recent years the Nike Sportswear business unit has stepped up its game and drops new limited edition styles on a monthly basis into carefully selected boutiques in New York, LA, Amsterdam, London, Shanghai, or other global lifestyle hot spots.
As a matter of quality, I think the two companies have roughly equivalent commitments to excellence. I can't say the same for Adidas' Reebok subsidiary - I don't respect Reebok products nearly as much - but the Adidas branded product is very good. Nike has invested many millions of dollars in catching up to the one category where Adidas had a big lead on them: football (soccer) boots, and Nike football boots are now easily equal or superior to Adidas football boots in any respect (depending, of course. upon individual preferences for fit and comfort).
In the running and basketball categories, most observers would agree that Nike shoes tend to be superior at all price points, and Nike's enormous market share lead over Adidas in these categories would bear that out.
Culturally, Nike is an American company and Adidas, for all its global reach, is still very much a German company. Adidas expends immense amounts of money and energy suing shoe companies that put two or four stripes on the sides of their shoes, although they have reached a kind of detente with the five stripes of the K-Swiss brand.