The English Zurich/Guinness Premiership (new sponsor from 2005) is a much higher standard of play than the Tooheys New Cup and Telstra QLD Premier Rugby because the internationals play in the cup. The Wallabies don't play in the Tooheys New Cup.
The Heineken/H Cup is the premier European club comp, comparable to the Super 14. However, 24 club sides play in the HC, so the talent is much more widely spread out, and the best players don't get to play if their club doesn't place high enough. For example, of the 12 Zurich clubs, only 6 (Sale, Leeds, Leicester, Saracens, Bath, and Wasps) play in the HC. Some of the world's best RU players (like Carlos Spencer and Andrew Mehrtens) won't be in the HC because their clubs didn't qualify. Mehrts won't even be in the Guinness Premiership, because the club he signed for, Harlequins, has been relegated to the Second Division (I forget the proper name). Jonny Wilkinson isn't in the HC either, because Newcastle Falcons didn't qualify.
I would say that the difference between European and SANZAR rugby would probably be comparable to the difference between ESL and the NRL. So Dan Lyle playing for Bath would be like he played for Hull FC.
You haven't heard of Dan Lyle because he doesn't play for a major nation that Australia would compete with, so unless you are in the habit of looking up USA Eagles or Canada scores, you wouldn't have heard of any North American rugby players.
And the games really aren't different, Union and League. Lyle's conversion is like an American footballer converting to play Lock in RL.