Using pg.connect is the way to go in a web environment.
PostgreSQL server can only handle 1 query at a time per connection. That means if you have 1 global new pg.Client() connected to your backend your entire app is bottleknecked based on how fast postgres can respond to queries. It literally will line everything up, queuing each query. Yeah, it's async and so that's alright...but wouldn't you rather multiply your throughput by 10x? Use pg.connect set the pg.defaults.poolSize to something sane (we do 25-100, not sure the right number yet).
new pg.Client is for when you know what you're doing. When you need a single long lived client for some reason or need to very carefully control the life-cycle. A good example of this is when using LISTEN/NOTIFY. The listening client needs to be around and connected and not shared so it can properly handle NOTIFY messages. Other example would be when opening up a 1-off client to kill some hung stuff or in command line scripts.