Aside from Frosty himself, the only other returning character from Frosty the Snowman (1969) is the traffic cop.
The design for Jack Frost would later inspire the design of the titular character in Jack Frost (1979). Rankin Bass regular Paul Frees provided the voice for Jack in Frosty's Winter Wonderland (1976) and would reprise the same role in Rudolph and Frosty's Christmas in July (1979). However, Robert Morse would give Jack his voice in Jack Frost (1979).
Jimmy Durante famously narrated Frosty the Snowman (1969) and sung one of the most memorable renditions of the song that served as the TV special's basis. However, although he was still alive during the production of Frosty's Winter Wonderland (1976), he was 83 and had retired from performing after having suffered a stroke that left him wheelchair-bound. Andy Griffith took his place.
The special first aired on CBS in 1976 but the rights are owned by Warner Brothers, and in 2010 Warner Brothers had full rights to the special which is why you don't see the first two Frosty specials on the same network in the USA.
In 2023 the special was released on DVD for the first time packed with the original 1969 Frosty the Snowman special. Previously the special was only available by itself or included in other Christmas special collections. The reason for the original special not including its official sequel for so many years is due to rights issues between two different companies owning each special.