Phoebe asks Ross if he's ever chased somebody through an airport before when he's going to tell Rachel he loves her. He replies sarcastically, implying 'no' but he actually did the same exact thing earlier in the series with Emily.
When Phoebe and Ross are buying tickets, there is a family of extras in line behind them. In the very next scene where you see Rachel looking for her boarding pass you can see the same family of extras. Moments later you find out that Ross and Rachel are at different airports.
On Newark airport you can see the aircraft Rachel is boarding, the gate is connected to the right-hand side to the aircraft. However, the passenger doors on passenger aircraft are normally always on the left hand-side.
When Ross and Phoebe go to see Rachel at JFK airport, they make a big point of having to stop and buy tickets so they have boarding passes to get through security. However, when they go to Newark airport they make it all the way to the gate but are blocked from seeing Rachel because they don't have boarding passes. Post-911, all airports require boarding passes to get through security, but it is never explained how Ross and Phoebe slip past security in Newark without them.
When Rachel attempts to board the plane the first time, the man at the desk looks at her passport upside down. Also, if you pause it here, you can see that the passport picture is a woman with dark, wavy hair, as though Rachel might have Monica's passport.