How is Zazen different from Anapanasati?

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.

They're the same practice. What you are describing are the generalities of Soto Zen. There's plenty of Zen that isn't Soto and that take a much more relaxed approach to posture, mudras, and the seat of concentration. In my own lineage of Obaku Zen, for instance, we're instructed to focus our attention on the tip of the nose and don't use any mudra at all. We also close our eyes.

Zen folk reach jhanas. If you are practicing samatha meditation, this is an inevitability just as if you were to take up running you'd inevitably develop improved cardiovascular capacity. The main difference is that we don't really care about which jhana we reach. We're not taught to, say, drop sustained and applied thought so as to enter into the second jhana. We just sit, focus on the breath, deal with what comes up, and figure it out as we go along. This emphasis on self-discovery, I'd say, is what truly distinguishes Zen from Theravada and Vajrayana practice. We try to find the path the same way that the Buddha did - through our own explorations. Our teachers don't instruct us. Instead, they keep us from going off the rails completely. They even purposely let us get lost from time to time (hello koans!). Obviously this is hard. Very hard. Ultimately, however, our realization is distinctly and utterly our own. It's not a parroted version of something we once read in a sutra somewhere. We find Buddha in our very bones.

Obviously, there's also that linguistic thing - jhana = dhyana = ch'an = zen. They're all the same word. But all that really isn't important. To the student of zen, it's what you do with that jhana that matters. Jhana that is not applied is self indulgence. Emptiness has to return to form in order for insight, and ultimately liberation, to occur.