Anapanasati and Satipatthana are essentially
the same thing. The Anapanasati Sutta says: "mindfulness with breathing, developed and cultivated, fulfils the four establishments/foundations of mindfulness".
However, in the scriptures, there are two scriptures with some different respective explanations, namely, the Satipatthana Suttas (MN 10 & DN 22) and the Anapanasati Sutta (MN 118).
The instruction in the Anapanasati Sutta (MN 118) is repeated in many other suttas, such as the excellent MN 62 and in most of the 20 suttas in the Ānāpāna Saṃyutta. The Ānāpāna Saṃyutta says in a number places (such as SN 54.8 & SN 54.11) the Buddha practised Anapanasati.
The Satipatthana Sutta is merely a list of various dhammas often illogically compiled together. It appears not possible the Buddha spoke the Satipatthana Sutta for various reasons, such as:
Including the Five Hindrances in the 4th satipatthana (because the hindrances are abandoned prior to the start of practice)
Including painful feelings in the 2nd satipatthana (because pleasant feelings rather than painful feelings arise from the prior calming of the breathing) and;
Referring to the impermanence of experiencing the 4 noble truths in the 4th satipatthana (because an enlightened person has a permanent discernment of the 4 noble truths).
In short, it appears the Satipatthana Sutta does not follow the right sequence of cause & effect as explained in AN 5.159 as the right way to teach the Dhamma.
Bhikkhu Sujato appears to say the Satipatthana Sutta was a fraud (Piltdown), as follows:
Some of the recent Burmese recensions have re-incorporated this entire section from the Dīgha Nikāya back into the Majjhima Nikāya, and even acknowledge this provenance by re-titling it the ‘Mahā Satipaṭṭhāna Sutta’. Perhaps a better title would be the ‘Piltdown Sutta’. This canonical innovation is extraordinary. While it is common for a word or phrase to slip between the cracks, I don’t know
any other place where a large body of text has been moved, obviously in
fairly recent times. No doubt this editorial innovation was designed to
further exaggerate the already excessive status of the Satipaṭṭhāna Sutta.
But the result is rather the reverse—such clumsy mishandling leaves alltoo-obvious fingerprints at the scene of the crime.
Bhikkhu Buddhadasa said:
Another common problem is that some people cling to and are stuck on the word satipatthana (foundations of mindfulness) far too much. Some go so far as to think that Anapanasati has nothing to do with the four foundations of mindfulness. Some even reject Anapanasati out of hand. In some places they really hang onto the word "satipatthana." They cling to the satipatthana of the Digha-nikaya (Long Discourses) which is not anything more than a long list of names, a lengthy catalogue of sets of dhammas. Although there are whole bunches of dhammas, no way of practice is given or explained there. This is what is generally taken to be satipatthana. Then it is adjusted and rearranged into these and those practices, which become new systems that are called satipatthana practices or meditation.
Then, the followers of such techniques deny, or even despise, the Anapanasati approach, asserting that it is not satipatthana. In truth, Anapanasati is the heart of satipatthana, the heart of all four foundations of mindfulness. The 16 Steps is a straight-forward and clear practice, not just a list of names or dhammas like in the Mahasatipatthana Sutta (Digha-nikaya#22**). Therefore, let us not fall into the misunderstanding that Anapanasati is not satipatthana, otherwise we might lose interest in it thinking that it is wrong. Unfortunately, this misunderstanding is common. Let us reiterate that Anapanasati is the heart of all four satipatthana in a form that can be readily practiced.
We have taken time to consider the words "satipatthana" and "Anapanasati" for the sake of ending any misunderstandings that might lead to a narrow-minded lack of consideration for what others are practicing. So please understand correctly that whether we call it satipatthana or Anapanasati there are only four matters of importance: kaya, vedana, citta, and Dhamma. However, in the Mahasatipatthana Sutta there's no explanation of how to practice these four things. It gives only the names of dhammas and expands upon them. For example, the matter of kaya (body) is spread out over corpse meditations, sati-sampajanna in daily activities, the postures, and others more than can be remembered. It merely catalogues groups of dhammas under the four areas of study.
The Anapanasati Sutta, on the other hand, shows how to practice the four foundations in a systematic progression that ends with emancipation from all dukkha. The sixteen steps work through the four foundations, each one developing upon the previous, and supporting the next. Practice all sixteen steps fully and the heart of the satipatthana arises perfectly. In short, the Satipatthana Suttas are only lists of names. The Anapanasati Sutta clearly shows how to practice the four foundations without anything extra or surplus. It does not mention unrelated matters.
The Anapanasati Sutta properly describes the true fruition or sequence of practise of a stream-enterer, namely:
The nature & causal relevance of the breathing is known & calmed.
From the calming of breathing, pleasant feelings arise, their causal nature is known & calmed.
From the calming of feelings, the mind and its purity or impurity is clearly known, any impurity is cleansed so the mind becomes pure & free.
Due to having a clear lucid mind, impermanence, unsatisfactoriness & not-self are fully (100%) experienced, which results is dispassion, the cessation of suffering & delusion and the giving up of regarding things as self.
In addition, the 7 factors of enlightenment are also explained in proper sequence at the end of the Anapanasati Sutta, including, most importantly, how to properly practise them, relying on "letting go/relinquishment" as the foundation.
As for the four noble truths & noble eightfold path, these must be understood intellectually prior to practise and they are realised gradually throughout each stage of practise. This is why each of 14 steps in the Anapanasati Sutta includes the phrase: "He trains himself" prior to them. "He trains himself" means training in the three trainings of higher morality, higher mind & higher wisdom, which includes the 4 noble truths.
In conclusion, the Satipatthana Suttas were probably composed after the Buddha for Buddhists who practise meditation less frequently; providing various different practises they can use.
Where as the Anapanasati Sutta (MN 118) has a very auspicious opening narrative, referring to famous disciples and the four levels of enlightenment in the Sangha. The Anapanasati Sutta is for serious practitioners who are practicing meditation for many hours per day.
In SN 54.11, it is said:
For whatever one rightly speaking would call, ‘a noble dwelling,’ ‘a divine dwelling,’ ‘a Tathāgata's dwelling,’ it would be the concentration of mindfulness with breathing that he, speaking rightly, would call, ‘a noble dwelling,’ ‘a divine dwelling,’ ‘a Tathāgata's dwelling.’”