Using the recumbent bike is a lot more comfortable but it also feels a lot more "lazy".
The primary advantage is you won't be putting any pressure on your genitals, causing them to go numb. This will depend on how upright you're sitting and on the seat as well, but having had my crotch go numb once, followed by a pins and needles feeling as blood got back in I just don't use an upright bike anymore.
You'll likely work your core slightly more on an upright bike.
Upright bikes require more core engagement to support your upper body. I qualified it with "likely" and "slightly" because that level of engagement is going to depend on how you personally ride the bike. If you are leaning heavily on your arms you are going to be taking more strain off your core for support as opposed to someone who is riding in a more neutral position. The seat in a recumbent bike provides the core support isolating the legs from the upper body more completely.
You'll find that they let you utilize two different groups of muscles.