This is an old debate on whether or to what extent the undergraduate syllabus should be related to research, or reflect, and stimulate discussion on, the nature of mathematics, as understood by professionals. J.E.Littlewood in his book "A Mathematicians Miscellany" claimed his excellent result at Cambridge was little relevant to research. Many, including me, have found the transition from undergraduate to postgraduate a culture shock. I was fortunate to find eventually that my way into research was in writing, and then getting a good problem.
Some have even found that their undergraduate degree "taught me to hate mathematics" or that "the difficulty and inaccessability of the courses left me and my friends scarred". See also a little article Carpentry.
Texts in algebraic topology often ignore relevant work on groupoids developed since 1967: see this mathoverflow discussion.
Another area neglected is fractals; I believe every undergraduate should be given some information on the Hausdorff metric and its application to fractals, since the latter have had wide publicity, and the course on this can be lots of fun with wide applications, provided it emphasises in the first instance ideas rather than proofs.
On the other hand, undergraduate courses often fail to give any background or context to the study of our subject. So the linked article on The Methodology of Mathematics has just been revised and republished. There are more articles on my Popularisation and Teaching page.
I confess to have been shocked that our students usually did not know why the angle in semicircle is a right angle, and so did some Euclidean Geometry for a special course, introducing proof as having the purpose of showing something surprising and capable of development, so showing the intent of a theorem.