Why does airspeed change when I adjust the prop?

When I change the prop setting on my (simulated) Baron 58, lowering
the prop RPM, my airspeed drops. I thought that for a given throttle
setting, the actual thrust produced by the powerplant was supposed to
remain the same for a wide range of prop settings, because of
automatic pitch changes made when I change the prop RPM. However,
that doesn't seem to be the case. A lowering of the prop RPM also
lowers airspeed, which implies a change in thrust. The fuel flow also
diminishes, which implies a change in power (?).

So, exactly what do I gain or lose by adjusting prop RPM when I'm
cruising along? Why would I want to change it? Some sources I've
read say that the prop makes less noise, which is surely true, but it
seems that I can't lower the RPM without losing airspeed (and thus I
must be losing power, right?).

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