It's somewhat dependent exactly what you're trying to do, and what database it is, but I'd say it's a lot friendlier in general to have just one connection.
If you can do what you want to do with just one connection, then you should use just one. You'll get better performance and use fewer resources.
This is especially true if the database server is non-local, or if it's one which takes a relatively long time to connect to (Oracle, maybe?)
In some databases there's a limit to the number of open objects per connection, or a limit to what you can do with a single connection - this is the only case where you'd reasonably want more than one.
If you do create more than one connection, take special care to ensure that you do things in a way which doesn't break in some cases.
Multiple connections can't see each others' transactions or temporary tables - so you must be aware exactly which connection you're using at any given time so that you don't create undesirable behaviour.
Mark
