As a graduate student, I found the old books of differential geometry used a different set of notation from modern textbooks. For example, Chern and Milnor defined the curvature 2-form by $d\omega-\omega\wedge \omega$, because they assumed that $Ds_{\alpha}=\sum_{i, \beta}\Gamma^{\beta}_{\alpha i}du_{i}\otimes s_{\beta}, \omega^{\beta}_{\alpha}=\sum_{i}\Gamma^{\beta}_{ai}du^{i}$, hence their connection matrix $\omega$ is the transpose of what we used nowadays. In today's notation, we have $\Omega=d\omega+\omega\wedge \omega$ instead.

This notation transition is not so difficult, until one actually encounters calculations using the two sets of notation, and the "translation process" is not easy. For example, the torsion free condition $\nabla_{X}-\nabla_{Y}-\nabla_{[X,Y]}=0$ is equivalent to $d\theta^{i}-\theta^{j}\wedge \theta^{i}_{j}=0$ using a local coframe field and connection matrix $\theta^{i}_{j}$. While this may be well-known to experts in the field, it certainly caused confusion to a new guy like me when I tried to read old papers (for example, Chern's paper on Chern-Gauss-Bonnet theorem). When I was learning differential geometry, "modern style" textbooks (De Carmo, Taubes, John Lee, Jurgen Jost, etc.) seem contented not to introduce the old notation. This problem can be solved by extensive googling around and reading old-style textbooks (like Chern's Lectures on differential geometry), but it takes some time at least.

I want to ask historically, what is the reason for this notation change? If I am not mistaken, this is a change of notation as well as a change of philosophy. In Chern's textbook there are a lot of explicit complicated computations to prove a trivial result (like connection exists on any manifold), and one gets the feeling that differential geometry is close to some kind tensor analysis. But I doubt if any modern reader (say, of John Lee's book) will feel the same way. What has happened since 60s-70s?

Reference:

Milnor: Morse Theory, Characteristic Classes (appendix)

Chern: Lectures on Differential Geometry, Chapter 4-5.