First, the difference between “表示” and “表达”:
Generally speaking, when translated as “express”, “表示” is more objective than “表达”.
“表示” does not have personal feelings or emotions, but “表达” is used specifically referring to people’s feelings.
Eg.
他对别的人从来不表示同情。
You cannot really use “表达” in this sentence, because it is a third party view saying that he does not show sympathy to others. However, you can use “表达” in this way:
他表达了深切的同情。(He expressed deep sympathy.)
In this sentence, “表达” is used directly after “him” to refer to his feeling. Thus, using “表示” in this sentence would be inappropriate.
Also, “表示” can mean other meanings except of “express”, for instance, it can mean “show”:
他手舞足蹈,表示他很高兴。(He is dancing, which shows/means he is very happy.)
You can see “表示” is used in a very objective way stating that “he is happy”, and you would not use "表达" in this sentence.
(There is only one exception that you can use "表示" with emotion, and that is "appreciation/ thanks". eg. 他表示感谢/ 感激。He said thanks. / He appreciated it.)
Second, “落伍” and “过时”
“落伍”s subject is more likely people, while “过时”s subject is more likely objects. Occasionally, you may use “落伍” for subjects, but you can never use “过时” for people.