Vietnam and Japan both use the twelve earthly branches, in the same order, written with the same characters. Three of the animals still do not match.
The fourth sign is a cat, not a rabbit
The best-known difference is the fourth. Japan's 卯 is the rabbit (usagi). Vietnam's Mão is the cat.
Why it became a cat is unsettled. The explanation most often given is that the older Sino-Vietnamese reading of 卯 sounded close to mèo, the Vietnamese word for cat. That remains a hypothesis rather than a proven account, and other explanations survive alongside it.
The practical result: in some years, Vietnamese New Year decorations are full of cats — a sight that puzzles Japanese visitors.
The twelfth sign is a pig, except in Japan
The twelfth differs too. Vietnam's Hợi is the domestic pig. Japan's 亥 is inoshishi, the wild boar.
Here it is Japan that stands apart. China and Korea both read the twelfth sign as pig. The character 猪 means domestic pig in Chinese but wild boar in Japanese, and that shift in reading is the likeliest source of the divergence.
The second sign is cattle, but not the same cattle
For the second sign, Sửu, a Vietnamese speaker pictures a water buffalo. A Japanese speaker pictures ushi, a cow.
Related animals, different pictures. One is the black buffalo standing in a paddy field; the other is not.
The year turns on different days
This is the part most likely to cause trouble at work.
Japan left the lunar calendar in 1873 (Meiji 6), so its zodiac year changes on 1 January. Vietnam keeps the lunar calendar, so its zodiac year changes at Tết.
Tết falls between late January and mid-February. For all of January, then, the two countries are in different zodiac years. New Year cards prepared without allowing for this carry the wrong animal.
Why it is worth knowing
Mention to a Japanese colleague that your fourth sign is a cat, not a rabbit, and the reaction is almost always surprise. Very few know.
One system, borrowed from one cultural sphere, splitting at particular points and keeping particular things. The zodiac is the clearest place to see it.