Live View uses a completely different method of focusing. It's called Contrast Detect and the imaging sensor itself, which is exposed in Live View, is doing the focusing. It's the most accurate focus method on your camera... but slow, slow, slow! It's actually the same type of focusing method many point n shoot cameras use... and why people are so amazed how fast SLRs focus can be and one of the key reasons they upgrade to them.
The normal focusing method, as seen through the viewfinder (with Live View off), is called Phase Detection and uses the array of AF points you see indicated in the viewfinder.
In the VF focusing mode, you have a choice of One Shot, AI Servo, or AI Focus. One Shot is used for stationary subjects. AI Servo is used to track moving subjects.
AI Focus isn't really a focus mode at all... it's more like automation on top of automation. The camera is supposed to decide whether or not the subject is moving, then switch to the other AF mode that's appropriate. In the past I've experimented with it and found it has a delay, so I stopped using it and just switch back and forth between One Shot and AI Servo myself.
One Shot is more accurate than AI Servo. It focuses, locks on the subject, then stops and gives Focus Confirmation. You can use it with multiple AF points or a single, manually selected point. The latter is the most accurate if possible. If using a single point, the center AF point is the most accurate and quickest in all Canon cameras (some have other higher performance points elsewhere).
Because it locsk on and then stops, One Shot is usable with a "focus and recompose" method, if using a single AF point.
AI Servo is continuously updated focus, constantly adjusting to follow a moving subject. It never locks, so doesn't give Focus Confirmation. Same as One Shot, it's most accurate using a single AF point, so long as you are careful to keep that point right on the place you want the camera to focus. You have to maintain half press on the shutter release button continuously for AI Servo to keep tracking. It is not practical to use for a "focus and recompose" method of shooting.
Subjects coming directly toward the camera or going directly away from it are the hardest for it to maintain AI Servo focus on. A subject moving past you, side to side or at somewhat of a diagonal, is easier to track.
You also might want to check out Back Button Focusing if you will be shooting in AI Servo a lot. This is a popular sports shooters technique, but really is usable with any type of moving subject and makes it possible for AI Servo to be used with a "focus and recompose" method, too. Basically, separating AF from the shutter release button... reassigning it to a button on the back of the camera that's operated with your thumb... puts the photographer more in full control of what focus is doing.
Yes, lenses can make a noticeable difference too.... In general, USM lenses (also HSM in Sigma or USD in Tamron) are faster, more accurate and quieter focusing. In general, lenses with larger apertures that offer up more light to the AF sensors also can be faster. There are exceptions, though... for example, by design the 85/1.2L is slower focusing, to emphasize accuracy over speed. Accuracy is more critical with super shallow depth of field rendered by very large aperture lens such as that.
Poor quality filters can have a negative effect on auto focus. Converserly, a lens hood might help at times, keeping oblique light from striking the lens.