Are ‘lose time’ and ‘waste time’ different in meanings? If so, how does it use the first and the second in different contexts? If not, what is it preferable?
They’re very different, though there are some circumstances where both could apply.
“Lose time” is used when there is an urgency to do something or get somewhere by some particular deadline, and some hold-up happens. It is typically, though not necessarily, something outside the person’s control.
We lost half an hour looking for the passports.
“Waste time” does not necessarily imply a deadline: there may be one, but there need not. It tends to be used when the delay is by somebody’s action, rather than by an accident or bad luck.
He wasted time putting everything away before he would come.
When used without a possessive pronoun (“wasting time”) it implies that somebody is doing something that makes themselves late (and often others as well). With a pronoun, we can talk about “wasting my time” or “wasting your time”.
When the latter form is used reflexively, as in “You’re wasting your time”, the implication is that you are doing something that you think is worthwhile but I’m telling you that it is not.