I used Bott and Tu as an undergrad and used Lee's book as a graduate student. Both books are equally rigorous and are more than sufficient for covering a semesters worth of differential topology and (if this is important to you) enough to prepare anyone for a qualifying exam in the subjects.

I personally prefer Lee's book, but if the comments are any evidence, this is not necessarily everyone's opinion. I'll add that I'm an algebraist (see: flair) and came into this course with an algebraic geometry background. Lee is chock full of great illustrations that clearly describe what's going on. The formulas that articulate what the photo shows are often extremely long and difficult to parse, but IMHO that is a symptom of differential manifolds and not a symptom of Lee. If you don't allow people to encapsulate information in functors/sheaves, then you're necessarily going to drown in notation, as functors/sheaves encode all the complex notation into themselves so you can "ignore" it until you actually do a computation, where the complexity will show itself all the same.

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Bott and Tu's book lacks exposition and the pictures aren't that great, which makes it hard to really see what's going on. In particular, it is a really good book for people who already know and are comfortable with some amount of differential topology/geometry and have an extremely strong grasp on linear algebra. This makes it a book beloved by professors, but as a student (especially a sophomore in undergrad) I found it impregnable.

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Some of this may be due to the fact that I worked through Tu first (and was mildly underprepared) then took a course with Lee where I was slightly overprepared. That being said, I do like how Lee works and find the exposition long but pretty important to my understanding.