The documentation for appendZoneOrOffsetId() says:

Note that this method is identical to appendZoneId() except in the mechanism used to obtain the zone.

The differences in the mechanism used to obtain the zone are as follows:

• appendZoneId:
  “the zone is obtained using a mechanism equivalent to querying the temporal with TemporalQueries.zoneId()”
• appendZoneOrOffsetId:
  “the zone is obtained using a mechanism equivalent to querying the temporal with TemporalQueries.zone()”

In practice, whether you will see a difference depends on what kind of datetime you are trying to format – whether it contains a zone id or just an offset. Consider the following code:

public static void main(String[] args) {
    DateTimeFormatter dtfZ = new DateTimeFormatterBuilder().appendZoneId().toFormatter();
    DateTimeFormatter dtfZO = new DateTimeFormatterBuilder().appendZoneOrOffsetId().toFormatter();
    tryPrint(ZonedDateTime.now(), dtfZ, "appendZoneId");
    tryPrint(ZonedDateTime.now(), dtfZO, "appendZoneOrOffsetId");
    tryPrint(OffsetDateTime.now(), dtfZ, "appendZoneId");
    tryPrint(OffsetDateTime.now(), dtfZO, "appendZoneOrOffsetId");
}

private static void tryPrint(TemporalAccessor dt, DateTimeFormatter dtf, String description) {
    System.out.printf("Trying to format %s using %s:%nt", dt, description);
    try {
        System.out.println(dtf.format(dt));
    } catch (DateTimeException e) {
        System.out.println(e);
    }
    System.out.println();
}

Output:

Trying to format 2024-09-24T13:08:10.042968898+01:00[Europe/London] using appendZoneId:
    Europe/London

Trying to format 2024-09-24T13:08:10.045608600+01:00[Europe/London] using appendZoneOrOffsetId:
    Europe/London

Trying to format 2024-09-24T13:08:10.046282182+01:00 using appendZoneId:
    java.time.DateTimeException: Unable to extract ZoneId from temporal 2024-09-24T13:08:10.046282182+01:00

Trying to format 2024-09-24T13:08:10.046858510+01:00 using appendZoneOrOffsetId:
    +01:00

As you can see, if you want to format a datetime with a zone region, then you can use either. But if you only have an offset instead of a region, then you need to use appendZoneOrOffsetId.

To complicate things further, there is a third method: appendZoneRegionId(). This behaves in a similar way to appendZoneId, except when you try to format a ZonedDateTime whose ZoneId is a ZoneOffset. A formatter with appendZoneId will happily append the offset if there is no region, whereas a formatter with appendZoneRegionId will throw an exception if the ZoneId is not a ZoneRegion:

DateTimeFormatter dtfZ = new DateTimeFormatterBuilder().appendZoneId().toFormatter();
DateTimeFormatter dtfZR = new DateTimeFormatterBuilder().appendZoneRegionId().toFormatter();

dtfZ.format(ZonedDateTime.now(ZoneId.of("Europe/London"))); // ok
dtfZ.format(ZonedDateTime.now(ZoneOffset.ofHours(1)));      // ok

dtfZR.format(ZonedDateTime.now(ZoneId.of("Europe/London"))); // ok
dtfZR.format(ZonedDateTime.now(ZoneOffset.ofHours(1)));      // exception!