The attached graph shows the average points for the Eastern and Western conference teams during the regular season. Of course, some years the conferences didn't have the same amount of teams but I think the graph shows the relative strength reasonably well. As can be seen, since the 99/00 season the Western conference teams have every season, on average, collected more points than the Eastern conference teams.
That's exactly what I recall from memory as well. When I started watching in the mid-90's, I felt the East was generally a little stronger, and by about 2001 things had shifted the other direction.
I remember the yearly dogfights between Edmonton, Phoenix, LA, Vancouver, and usually another team or two for the final two or three playoff positions during the 2000-2004 span. Great hockey down the stretch as you could move from 10th to 6th with a single win, or vice versa. It was basically playoff hockey from March 1st onward. I remember one year the Oilers went something like 12-3 over their final 15 games
and it wasn't enough to overcome a gap that was never larger than two or three points.
On one hand it made for awesome regular seasons, but if you were one of the teams left out in the cold at the end, it annoyed you knowing that you probably would have been 5th or 6th in the East those years. The 2002 Hurricanes, to pick on somebody, won their division and advanced to the final, and wouldn't have made the playoffs in the West.