How much does the Spotify app use the network?

"Being a P2P application, it will be using network traffic when:

You are requesting data from peers
Peers are requesting data from you

so even if you aren't listening yourself, you could be uploading to other users.

To quote the Spotify wiki article:

The contents of each client's cache is
summarized in an index which is sent
to the Spotify stream hub upon
connecting to the service. This index
is then used to inform other clients
about additional peers they can
connect to for fetching streamed data
for individual tracks being played.
This is accommodated by each client,
upon startup, acting as a server
listening for incoming connections
from other Spotify users, as well as
intuitively connecting to other users
to exchange cached data as
appropriate. There are currently no
official details from the developers
about how many connections and how
much of a user's upstream bandwidth
the Spotify client will use when
streaming to other users; the Spotify
client offers no way for the user to
configure this, but this may change in
the future.
Audio streams are in the Vorbis format
at q5 (approx ~160 kbit/s), or
optional q9 (approx ~320kbit/s)
for premium subscribers, the highest
streaming rate for any online service."