How close are planes allowed to fly next to each other?
Your plane was at least 300 meters above the aircraft you saw.

The limits are determined by the altitude and capabilities of the aircraft.


Up to 29,000 ft - 300 meters vertical separation
29,000 ft and above - 600 meters
29,000 - 41,000 ft (where there is RVSM (Reduced Vertical Spacing Minima)), the minimum separation is 300 meters, but the aircraft have to be equipped with specific instruments. Most modern airliners are equipped to fly in RVSM approved airspace.
41,000 feet - 60,000 ft, 300 meter RVSM
60,000 feet and above, 1.5 KM irrespective of RVSM


You won't feel turbulence unless you are flying behind an airplane, or you fly into an airplane's wake.  Turbulence happens in disturbed air and this is usually behind the airplane's wingtips.

All modern airliners are equipped with TCAS (Traffic Collision Avoidance System); which consist of visual indicators on the relative position of nearby aircraft.

Here is an image from flightgear wiki that shows how this looks on some modern airliners: