Once upon a time, Google dared to experiment with HTTPS encryption for their search instead of allowing all search data to go unencrypted through the wire. For this experiment, they created a new subdomain: encrypted.google.com was the address where your could get some extra privacy. What some people apparently didn’t notice: the experiment was successful, and Google rolled out HTTPS encryption to all of their domains. I don’t know why encrypted.google.com is still around, but there doesn’t seem to be anything special about it any more. Which doesn’t stop some people from imagining that there is.
Myth #1: Your data is extra private, thanks to the extra encryption
The “encrypted” in “encrypted.google.com” refers to HTTPS encryption. You get it on any Google domain, and pretty much every other search engine switched to HTTPS-only as well. There is no indication that Google experiments with any other kind of encryption on this domain.
Myth #2: encrypted.google.com doesn’t send your search query to websites you click on
That myth seems to be based on this answer by a Google employee noting differences in referrer handling. And maybe in 2013 it was even accurate. But these days anybody running a website will confirm that they don’t see your search query, no matter what Google domain you are on. Until recently the reason was Google’s Instant Search feature, the search terms simply weren’t part of the address. Now Instant Search is gone, but Google uses Referrer Policy to prevent giving away your search terms to other websites.
I did notice a few actual differences on that domain. For example, encrypted.google.com won’t redirect me to google.de because of my geographical location — but it will still show me results that are specific to Germany, same as google.de. It also doesn’t provide links to some other Google apps such as Maps. Not really a reason to prefer it over regular Google domains, but maybe somebody knows about more substantial differences and can list those in the comments.