How staying in the present moment(here and now) is different from awareness and mindfulness, and what according to you is better?
which is better staying in present moment or being aware?
To counterpose such sentences they should have particular definitions.
I think that "staying in the present moment" is used as a synonym to "being aware", because minds of people usually aren't collected, i.e. they are scattered, distracted. They tend to be involved in repetitive thoughts about past mistakes or joys, future plans, anticipations and fears, etc.
For example, when you are eating, you are in scattered thoughts rather than here; when you are walking, you want to be at the destination already. So the physical body is walking, but mentally you are not quite here.
Perhaps it isn't compatible with living truly happily, because if you want to be happy (or to be in nirvana), it would mean you would be living through this very moment, with satisfaction. Otherwise you just waste the time of your life, throwing moment after moment away.
In nirvana you would use — live through — every moment with complete dedication, completeness; not escaping the present into thoughts, daydreaming etc.
Thus focus on this real perception, let your mind be collected: that's a starting point for actual awareness and mindfulness. That's re-educating your mind, pulling it back from habits of being unaware or half-aware.
Good householder "Son of liberty",
might help, once on the path section of concentration.
Being, aware, actually means eating, eating off, burning away. If that isn't done in higher spheres, then it's just eating of ones merits.
As for the place where he might be: sati, mindfulness toward ones duties in ones relation, toward Sila, toward Generosity, toward reflecting backwards.
Keeping precepts in mind, train right there, will make the further way accessible by itself (given causes).
And right concentration has right mindfulness (remembering, keeping in mind of what is kusala) as it's prerequisite.
(And pull the hands out of your trousers, and stand straight, as such is a good sign that one prefers eating off past merits over the sati of a son of liberty)
[Note that this isn't given for stacks, exchange, other world-binding trades and gains, wrong concentration, but for exit from this wheel]
In Buddhism, the official terminology is mindfulness (sati) & clear comprehenshion/ready wisdom (sampajjana).
Mindfulness means 'to remember' or 'keep in mind'. Sampajanna refers to whatever wisdom or right view is required in relation to a specific situation.
In summary, mindfulness remembers to keep sampajanna (ready/situational wisdom or clear comprehension) in the mind.
Staying in the present moment (here and now) or samadhi (concentration) can be a result of mindfulness & clear-comprehension, as described in the sutta MN 131.
Alternately, staying in the present moment without wisdom can be a type of concentration that is both wrong and short-lived. If there is no right mindfulness & clear-comprehension, concentration in the present moment cannot last long nor can develop into genuine samadhi.