Philips Hue: You can use the phone app to control individual bulbs directly with bluetooth, but for more control you need also the Hue brigde, which talks to the bulbs over Zigbee (a wireless protocol), and to your phone app over your wifi. For initial setup, the bridge needs an internet connection, but otherwise it will work as long as your home wifi is up, even if there is no connection to outside internet. The features (timed schedules, scenes) live in the Hue brigde, so everything works also without outside internet.
LIFX: No Zigbee, no bridge. These bulbs connect to your 2.4GHz home wifi. More advanced features (schedules, scenes) live in the LIFX company server, so won't work without internet connection. But without outside internet connection, you can still use the phone app via your wifi router to control the color and brightness of individual bulbs. If you're a home automation hobbyist, you can give color and brightness commands to LIFX bulbs over your 2.4Ghz wifi, so you can program your own timed schedules that would work without outside internet connection.
Other smart bulbs: Cheaper. Philips Hue and LIFX are the two most expensive.
I understand LIFX is best for bright colors. But Philips Hue bulbs don't contain just RGB leds, but also leds for white and warn white, so the bulb may have 5 different types of leds. So Philips might produce better near-white "natural" light than same colors produced by combining only red,green,blue leds. Not sure if this matters to everyone or only to some lighting connoisseurs.