A couple of years ago, I was in the world's largest Jewish city around the time of Succoth, the autumn Feast of Tabernacles. Passing through Grand Central on my way to visit the Frick Collection, I was accosted by a charming young man, with skull-cap, who seemed all of eight years old, who profferred me a strip of palm with the question "You're Jewish?" It took me but a nanosecond to decide that this was not the occasion to offer subtle distinctions; so I just said "I'm afraid I'm not", at which he cheerfully withdrew the palm and passed on his way. Neither of us attempted to proselytise the other.
Here are some words from Fr Thurston's admirable CTS pamphlet dated 1949:
It is perhaps sometimes forgotten that the association of the cry Hosanna with the waving of palm-branches does not date merely from our Lord's solemn entry into Jerusalem. If the people saluted our Saviour in this manner at the moment of His triumph, it was because both action and words were familiar to them as part of the ceremonies of one of the most joyful festivals of the year. On each of the seven days of the feast of Tabernacles the people moved in procession about the altar in the court of the Temple, making their boughs of palm bend towards it, and shouting Hosanna ("save now"), while the trumpets sounded. Moreover it would seem that verses 25 and 26 of Psalm cxvii, beginning Hosanna and containing the phrase, "Blessed is he that cometh in the name of the Lord," were used as a sort of responsory to the great Hallel (Psalms cxii-cxvii), which was recited on this occasion. When it is added from the explicit tradition of the Talmud that the children who were old enough to wave the palm-branches were expected to take part in the celebration, and that the boughs themselves came in the course of time to be called Hosannas, it will be clear how close a connection there is between the Christian procession of Palm Sunday and the palm festival still observed by the Jews after the harvest in the autumn. Both the ceremonies of the Jews in their synagogues and our own procession on Palm Sunday represent a rite which has existed in some shape from the time of the entry into the promised land more than 3,000 years ago.