"When you feel the pressure" - is a phrase which implies that the pressure is winning. You can feel the stress and it's power, and this causes you to worry a little bit. For example, you are playing soccer against another team, you are ahead by one goal but they are really charging forward and getting lots of chances to score on your net. You have 3 more minutes to play and you must defend your net and not let them score. You're worried, you feel nervous, you feel the pressure! "The pressure is on" - is a phrase which implies you recognize there is pressure but you are able to deal with it. You may even enjoy this feeling. For example, you are playing soccer against another team and you are losing by one goal, but your team is playing really well and you think you may score in the last 3 minutes. You're excited, you feel confidant, you have to work hard but you know you can succeed when the pressure is on!

I think “ feel the pressure” is more a subjective expression, somebody feel pressure. For example, she feels pressure of her final exam. But “the pressure is on sb/sth” sounds more like a passive, it means the pressure is coming from somewhere else not your self. For example, the whole team put the pressure on him.