To "witness" means to:
"testify to; give or afford evidence of"
A "witness" is an:
"individual who, being present, personally sees or perceives a thing; a beholder, spectator, or eyewitness"
Therefore, "bearing false witness" means:
"testifying falsely to something that one has seen or perceived"
The key is that this is a legal definition - where someone is "testifying/witnessing" for or against someone else. In that setting, the reason for the "witness" is to establish a punishment or lack thereof for someone else - to judge as guilty, innocent or not guilty of something. Thus, bearing false witness causes an injustice to occur - either by removing a penalty that should be enforced or imposing a penalty that should not be enforced. In modern verbiage, it is perjury - and perjury is a crime specifically because of the effect it might cause on others and on society as a whole. It really is a serious action to subvert justice in a legal setting - either by assisting in the punishment of someone who is not guilty or the escape from punishment of someone who is.
A "lie", on the other hand, simply is a statement that is known to be false by the person making it. It can have nothing whatsoever to do with justice. It might be telling someone their hair or dress or shoes look fine - which might or might not be a good idea, depending on the person and the situation. It might be saying you are fine when someone asks - even if you feel lousy. It might be protecting someone from unnecessary harm, torture or death - which I can't see as a bad thing. It even might be a social convention that is understood to be meaningless by everyone who hears it.
I agree totally with the injunction that forbids bearing false witness; I try to avoid lying whenever possible, even if I have to be a bit evasive or ambiguous in some cases to not lie; I am totally fine with some lies. That's why I always answer, "I try my hardest," when I am asked the question in the temple recommend interview about being honest.