What are practical differences between kernel k-means and spectral clustering?
The differences are indeed not too large. There is a paper called
Kernel k-means, Spectral Clustering and Normalized Cuts by
Inderjit S. Dhillon, Yuqiang Guan, Brian Kulis from KDD 2004
that is discussing that relationship.
The authors show that spectral clustering of normalized cuts is a special case of weighted kernel k-means. The reason is that the &quot;graph cut problem and weighted kernel k-means problem can both be written as trace maximization problems&quot;.
They write that

this has important implications: a) eigenvector-based algorithms, which can be computationally prohibitive, are not essential for minimizing normalized cuts, b) various techniques, such as local search and acceleration schemes, maybe used to improve the quality as well as speed of kernel k-means.

Further they combine ideas from both to improve the total results and

show that using eigenvectors to initialize kernel k-means gives better initial and final objective function values and better clustering results.

Regarding robustness in spectral clustering you might want to look at Robust Spectral Clustering for Noisy Data by Aleksandar Bojchevski, Yves Matkovic, Stephan Günnemann, published at the ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD), 2017: https://www.kdd.org/kdd2017/papers/view/robust-spectral-clustering-for-noisy-data

Inderjit S. Dhillon, Yuqiang Guan, Brian Kulis, Kernel k-means: spectral clustering and normalized cuts, KDD '04: Proceedings of the tenth ACM SIGKDD international conference on Knowledge discovery and data mining, August 2004, Pages 551–556, https://doi.org/10.1145/1014052.1014118