Nath and BetaCAV have good descriptions, I think, though I'd like to point out something else.
Knight Errant started out as corporate security, generally recruited from ex-armed forces: "This is Ares property, please depart the premesis." They'll be polite to the customer until it is time for them to no longer be polite; then they'll shoot to kill. These are ex-Army and ex-Marines given courtesy training and well-defined parameters of how far they're supposed to allow 'the customer' to go, after which point they'll have no problem putting the now-perp down hard. They have a book they're trained to, and they'll stick to it very firmly, because they never know if they're going to get transfered from an Ares corporate facility to the back end of Podunk, IA. Everyone in a city with whom KE has a contract is a customer, until they cross the line; then they're a target. Taking prisoners is encouraged, but you can't guarantee a prisoner in a firefight. Your archetypical KE officer is young, fit, square-jawed, and ready to fire an SMG or assault rifle with speed and accuracy; depending on patrol area, they at least have one of these in the cabin of their patrol vehicle, if not brought with them in a combat-carry position. He spends his on-shift downtime at the "precinct house" in the gym with the rest of the team lifting weights.
Now, while Lone Star has a book that they're trained to as well, a large number of the people who were originally recruited into the company were (as has been pointed out) recently-discharged cops - which means the corporate culture is that of 'police officer'. Cops are trained to a much looser idea of offenses, as it were - instead of a binary 'friend/enemy' solution, it's more of a digital 'where you fall on the citizen/perp scale'. They aren't cops, but they tend to think like cops. They're almost always going to be providing security for a community, even if that community is only a gated collection of well-to-do houses; security for a corporate facility is waaaay down on their client list. Your archetypical LS officer is older, a little (or maybe a lot) out of shape, looks a little soft and/or friendly (well, if you're an upright human citizen), and will tend to go for the heavy pistol riding on his hip. He has a shotgun, sure, but it's in the trunk of his patrol car, and he only gets it out if he really has to. He spends his on-shift downtime in a social situation either with his compatriots (at a bar or in the precinct room) or the public (in a diner/restaurant/donut shop) shooting the bull and decompressing.
As Nath said - LS is more coplike; KE is more akin to a military force 'filling in' as cops. KE gets seen as being more professional, but what that actually tends to mean is that you get the big-brother vibe turned up a notch or two.
