As I understand it, the Bourbaki school was basically obsessed with the idea of making maths rigorous to the point where their published work is extremely difficult to understand and you basically already have to know the subject to read the stuff. Other people, Arnold in particular, felt that it was important to learning to gain a more intuitive understanding even if that meant sacrificing some rigour along the way, and that once you got there you could go back and sort of fill in the blanks.
That's why that wiki page says
Arnold was an outspoken critic of the trend towards high levels of abstraction in mathematics during the middle of the last century. He had very strong opinions on how this approach—which was most popularly implemented by the Bourbaki school in France—initially had a negative impact on French mathematical education, and then later on that of other countries as well.
So a pastiche of the Bourbaki type criticism of Arnold's works would be that they are elegant but skip over some of the foundational steps that are necessary to be fully rigourous and the equivalent critique of Bourbaki is that their stuff is dry, tedious and impossible to understand which just puts people off the subject entirely rather than teaching them.
Arnold is very influential especially in "Russian math" pedagogy/philosophy, which could be contrasted a bit with "French math" where Bourbaki was very influential. I think anyone interested enough to read the Bourbaki article might be interested in reading about him since his wikipedia page has a few references pointing to interesting discussions about Bourbaki's influence. I should have mentioned a little details in the original comment, sorry!
Also not to mention that he has written many great textbooks on top of his research work. I would recommend anyone in an intro or second mostly-methods-based DE course to read his Ordinary Differential Equations. He doesn't entirely avoid useful methods (as I recall) but the approach was extremely different to what I had seen before, but so natural (pointing at geometry and topology, immediately discussing vector fields and a nice notation for flows in the beginning chapter).
