Here are a few mentions:

The persistence of Neanderthal DNA in the modern human gene pool demonstrates adaptive introgression, the process of gaining beneficial genetic material from another species. For modern humans expanding into Eurasia, these borrowed genes offered a shortcut to adaptation. Acquiring pre-adapted genes was much faster than waiting for new mutations to arise and spread.

The advantages in the past were clear. A more aggressive immune response helped fend off local infections. Genes that promoted faster blood clotting were beneficial for surviving injuries from hunting or other daily hazards. Similarly, changes in skin pigmentation may have helped optimize vitamin D production in regions with less sunlight.

This ancient inheritance has led to an “evolutionary mismatch” in the modern world. Traits that were once advantageous can have neutral or negative consequences in current lifestyles. For instance, an immune system calibrated for constant pathogenic threats can become over-reactive in our sterile environments, contributing to allergies and autoimmune disorders.
https://biologyinsights.com/the-modern-neanderthal-genetic-traits-inherited-from-dna
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The good, the bad, and Neanderthalic immunity

Abstract
Introgression with archaic hominins and subsequent natural selection has shaped the immune system of modern humans. Recently, Sun et al. investigated the immunity advantages of a Neanderthalic variant in the membrane-bound immunoglobulin G1 (IGHG1) gene, activating pathogen-specific antibody production toward modern threats yet conversely increasing the risk of autoimmune diseases.
https://www.sciencedirect.com/science/article/abs/pii/S0168952524002610
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https://emedicine.medscape.com/article/330815-overview
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Originally posted Feb. 13, 2023; reposted in conjunction with publication of the PMR supplement to the February 2024 issue of The Rheumatologist.
PHILADELPHIA—Polymyalgia rheumatica (PMR) is a chronic inflammatory condition that almost exclusively affects individuals older than 50