What is the difference between these two varieties (if any)?
It appears that these are just different varieties, like red and golden delicious apples are different. The sugarcane jujube is smaller and sweeter than the other.
Here's a jujube fact sheet from California Rare Fruit Growers, Inc. that lists Sugar Cane:
Small to medium fruit which can be round to elongated. Extremely sweet fruit but on a very spiny plant. The fruit is worth the spines!
See also this article on growing jujube in Texas:
The better varieties for fresh use are 'Honey Jar,' 'Sugar Cane,' 'Li,' 'Shanxi Li,' 'Sherwood' and 'Chico' ('GI 7-62'). Of these the largest is 'Shanxi Li' and the smallest is 'Honey Jar.' 'Honey Jar' is the juiciest. 'Chico' ('GI 7-62') matures in mid to late season and 'Sherwood' is the last of these to ripen in early October. 'Li,' 'Shanxi Li,' 'Honey Jar' and 'Sugar Cane' ripen in August and early September.
See also this seller's page that makes it a bit more obvious that Sugar Cane is just a named cultivar.