Zazen and Ānāpānassati are different in the sense that may matter to those who are starting the Noble Eightfold Path. This takes a little explanation and some context.
The Buddha had two chief disciples, Sāriputta and Moggallāna:
MN141:5.1: Mendicants, you should cultivate friendship with Sāriputta and Moggallāna.
MN141:5.2: You should associate with Sāriputta and Moggallāna.
MN141:5.3: They’re astute, and they support their spiritual companions.
MN141:5.4: Sāriputta is just like the mother who gives birth,
MN141:5.5: while Moggallāna is like the one who raises the child.
MN141:5.6: Sāriputta guides people to the fruit of stream-entry, Moggallāna to the highest goal.
Each of these two great disciples taught in different yet complementary ways. Sariputta was a great teacher of the text. Moggallāna was a great teacher of insight. As students, we actually need both. We need to understand the text and we need guidance for insight.
Although both were adepts in depth and breadth, Ven. Sariputta would probably have been inclined to teach Ānāpānassati, while Ven. Moggallāna would probably have addressed direct experience and insight. Anapanasati is described in great detail in MN118, which is quite thorough and touches on many concepts explained elsewhere in the text. Sariputta would have answered all questions about Ānāpānassati and its terms:
MN118:15.2: Ānāpānassati, bhikkhave, bhāvitā bahulīkatā cattāro satipaṭṭhāne paripūreti.
MN118:15.2: Mindfulness of breathing, when developed and cultivated, fulfills the four kinds of mindfulness meditation.
In contrast, Ven. Moggallāna might have taught more in the style of Zen and zazen. Zen is terse and tough.
Venerable ones, those are only noisy names, wordy sentences,
and are all a mere change of robes. Names arise from the ocean
of breath in the region of the belly; their fierce drum beat rattles
your teeth so that they stutter out interpretations. Do you not
see that these are but illusory phantoms?
Personally I did things out of order. I started with Zen. But it only made sense to me after I read the suttas. In particular, MN118 was helpful regarding meditation on the breath.
Because starting with Zen was difficult for me, I'd recommend starting with at least a brief study of MN118.
However, the odd thing about the Noble Eightfold Path is that no matter where you start you always end up walking exactly as you need each step by step.
Basically, choose whichever method suits you best now.