Hi @M_F,
The biggest difference is that QIIME 2 is intended as an end-to-end analysis platform for microbiome data, while DESEQ(2) is focused on the specific task of differential abundance testing. DESEQ won't, for example, help you with quality control of your data (e.g., as you'd do with qiime dada2 denoise-paired) or taxonomic annotation (e.g., qiime feature-classifier ...). phyloseq provides more functionality than DESEQ, but as far as I understand it hasn't been actively developed in several years so methods may be outdated.
QIIME 2 provides some functionality for differential abundance testing - we currently recommend ANCOM-BC. See the documentation of that here and here. We're also working on adding support for ANCOM-BC2 in this release cycle (due out in April 2025, but development versions may be available sooner).
It's possible to export data from QIIME 2, such that you could run differential abundance testing with DESEQ2 if you'd like, but ANCOM-BC has been shown to have considerably lower false positive rates than DESEQ2, so I don't recommend that. (Update: this has come up on the forum here.) This paper provides a really nice comparison of methods for differential abundance testing (thanks for the link @jwdebelius!).
QIIME 2 does integrate support for multiple testing correction where applicable. For example, in the context of differential abundance testing, this is included in the output of QIIME 2's ANCOM-BC wrapper. For example, see the "q-val" in this screenshot, which is derived from this figure in second bit of documentation that I linked you to above:
Another big selling point for QIIME 2 is that it integrates support for complete analysis reproducibility through its provenance tracking and Provenance Replay functionality. This can be hugely helpful to you when you're preparing your results for publication, or when you need to document your work for others. Oh, and QIIME 2 has an awesome forum, with many awesome moderators who are here to help as you work through your awesome analysis! 
Do others have thoughts to share on this?