As @LPChip said, LEDs flicker at a high rate. This rate is equal to twice the frequency of your powerline (afaik, 50Hz in Europe, 60Hz in USA). If your exposure time is a few times the flicker rate, you'll have no trouble. Also possible is a exposure time of exactly a low multiple of the flicker frequency (so if your power is using 60 Hz, you can safely use 1/120 s or 1/60 s). For video, I'm not sure how to deal with the flicker, but a frame rate of exactly the power frequency might work. (if you use a framerate even slightly different from the net frequency, you'll catch a different phase of the light cycle at each image, so some frames will get full light, some will catch the full dark period, and the others will be somewhere in between.
The differences you noticed between different lamps can have a few causes.
One is the driver needed to transform the incoming alternating current into direct current. More expensive circuits can remove the pulsing that remains after the initial transformation of the alternating current.
Another might be related to the composition of the lamp: white leds are actually (blue?) LEDS within a (fluorescent) material that converts part of the LED light to longer wavelengths (green, yellow, and red light). If that material shows some phosphorescence, the lamp will continue emitting light for a short time (ms or µs) after the LED "switches off". That can diminish the flickering.