Both are geocoders using OpenStreetMap data and backed with Elasticsearch. The main difference is how they handle geographical context, eg to determine in which city/region each place is located. Which is useful both for indexing the data and producing human-readable labels.

Pelias is using its own gazeteer (https://whosonfirst.org/) to index locations and remarkable zones. Whereas Mimirsbrunn relies only on OSM data for this task using Cosmogony (https://github.com/osm-without-borders/cosmogony), a tool that extracts official administrative boundaries and other shapes from OSM data. But the world is a complicated place, and OSM tagging scheme doesn't always lend itself to geocoding purposes. So it comes with a lot a caveats which are still being working on :)