I agree with the other posters who have said that 6 hours is NOT enough time to go into Manhattan and still make it out to JFK in time for your international flight.

Generally, I tell people that 8+ hours layover would be doable, but not less than 8 hours. You should allow 45 minutes to 1 hour in each direction (assuming AirTrain->E, a little less time on AirTrain->LIRR to Penn Station), but anything could happen. And your LGA arrival means you will be forced to use some kind of non-rail transportation for part of the way, which will subject you to the unpredictability of NYC traffic.

So, in short, I think that you made the right decision just to go straight to JFK and wander around there. Definitely check out the shopping/dining mall in T4, and look around T1 as well. If you get off the AirTrain at T5 and walk on the long curving walkway toward JetBlue's new terminal, you can see the historic TWA terminal on the right.