I have see both used. I was first exposed to the usage of caudate at the 1995 SSAR meeting there. There were a number of formal and informal talks that used the word caudate which I think (since I am not a language specialist) is a slang usage. That said, it appears to me that caudate is used more frequently in the USA and urodele is more common elsewhere.
From Biology of Amphibians page 495, top left column, second sentence " The substitute ordinal name Urodela dating from Latreille 1825, commonly used in place of Caudata.
From Herpetology An Introductory Biology of Amphibians and Reptiles page 4, 1st column, second topic 1st paragraph, 10th sentence,
"The salamanders, labeled with the node-based name Caudata (= "having tail") and the stem-based name Urodela (= "tail visible"), have cylindrical bodies,......"
I am not the best at systematics so maybe someone else can explain the difference between node based and stem based if necessary.
Ed