If you use a single connection and share it, only one thread at a time can use it and the others will block, which will severely limit how much your application can get done. Using a connection pool means that the threads can have their own database connections and can make concurrent calls to the database server.
See the postgres documentation, "Chapter 10. Using the Driver in a Multithreaded or a Servlet Environment":
A problem with many JDBC drivers is that only one thread can use a
Connection at any one time --- otherwise a thread could send a query
while another one is receiving results, and this could cause severe
confusion.
The PostgreSQL™ JDBC driver is thread safe. Consequently, if your
application uses multiple threads then you do not have to worry about
complex algorithms to ensure that only one thread uses the database at
a time.
If a thread attempts to use the connection while another one is using
it, it will wait until the other thread has finished its current
operation. If the operation is a regular SQL statement, then the
operation consists of sending the statement and retrieving any
ResultSet (in full). If it is a fast-path call (e.g., reading a block
from a large object) then it consists of sending and retrieving the
respective data.
This is fine for applications and applets but can cause a performance
problem with servlets. If you have several threads performing queries
then each but one will pause. To solve this, you are advised to create
a pool of connections. When ever a thread needs to use the database,
it asks a manager class for a Connection object. The manager hands a
free connection to the thread and marks it as busy. If a free
connection is not available, it opens one. Once the thread has
finished using the connection, it returns it to the manager which can
then either close it or add it to the pool. The manager would also
check that the connection is still alive and remove it from the pool
if it is dead. The down side of a connection pool is that it increases
the load on the server because a new session is created for each
Connection object. It is up to you and your applications'
requirements.