I saw a comment in another post suggesting that these are two different things. Can anyone explain or refute?
Mindfulness ('sati') means 'to remember'. The breathing as a meditation object cannot be 'remembered'. Instead the breathing is observed, which is called 'anupassi'.
The purpose of 'mindfulness' in Buddhism is 'to remember' to keep the mind in a state of right view & right intention (refer to MN 117) or 'to remember' to keep the mind free from covetousness & distress (refer to MN 118; MN 10; etc).
Therefore, to practise 'mindfulness of breathing' is an impossibility because mindfulness (sati) means to remember rather than to observe (anupassi).
Instead, the practise is 'mindfulness with breathing', which means to remember to keep the mind free in right view free from craving, as instructed in the four noble truths. When this is properly done, awareness/observation of breathing will naturally arise due to the quiet & silence of the mind.
Thus, mindfulness with breathing is remembering to keep the mind free from craving with awareness of breathing arising automatically/non-volitionally as the 'sign' of this right mindfulness.
Those that try to practice mindfulness of breathing rarely progress because the mind must let go of craving to progress; which includes letting go of the craving or intention to watch the breathing.
Ideas about 'mindfulness of breathing' are yogic or Hindu meditation.
'Mindfulness with breathing' is Buddhist meditation.
Mindfulness is often mentioned within the scope of maintaining ethical behaviour in the actions of body, speech, and mind. However, many people conflate 'Mindfulness' with 'Vipassana'.
Mindfulness of mental activity is by far the most challenging, and likewise the most important.
Monitoring the breath is a good way of calming excessive internal chatter. Once you have the simple ability to monitor the breath, switch to monitoring the mind.
The effort you use in your meditation is: a mode of joy in having the opportunity to practice, the determination to practice, and faith in your ability to practice.
I believe that vipassana arises out of mindfulness: as we monitor our minds we begin to perceive the nature of mind - not through thinking about it, but from experiencing how it shapes our world. @Dhammadhatu or others may be able to point to text that support (or lay waste) to such an idea.