When to use a new variable vs string interpolation?

Your refactor seems to have reduced the readability of your code, which is usually the opposite of your intent when refactoring. The code may be terser, and string interpolation and re-use is a part of how you've achieved this, but it doesn't make the code easy to follow. That will make it harder for you and your colleagues to modify it over time.

I actually prefer your original code a lot. You've pulled two methods out of your main execution flow, so it's possible to see at a glance what the code is doing - validate a directory, find all the files in it and do some form of outputting on each. That clarity is missing from the newer code because it is all written at the same level.

Your code makes use of #dup and #clone a lot. These methods are not needed in many of the places you use them, so they add noise to the code. For example files = folder.clone + "*.flv" does not need the #clone call because the + method will create a new String from the content of folder.clone and *.flv. Writing files = folder + "*.flv" would work just fine.

Here is a take on your original code that adds some readability by avoiding repetition and giving fuller names to variables and methods.

    def normalize_directory(dir)
        normalized = dir.gsub('\\', '/')
        normalized += '/' unless normalized.[-1,1] == '/'
    end

    def validate_directory(dir)
        if dir.empty?
            puts "Enter the full directory path of the flv files." 
            dir = gets.chomp
        end
        dir = normalize_directory(dir)

        until File.directory?(dir) && !Dir.glob("#{dir}*.flv").empty?
            puts "That directory either doesn't exist or contains no .flv files. \nEnter the full directory path of the flv files."
            dir = gets.chomp
            dir = normalize_directory(dir)
        end

        dir
    end

    def output_html_wrapper(flv_filename, output_folder)
        html_filename = flv_filename.gsub(".flv", ".html")
        vid = flv_filename.slice(0..6)

        html_body = $EMBED.gsub("sample.flv", vid) 
        html_output = File.open(html_filename, "w")
        html_output.write(html_body)
        html_output.close

        link = vid.gsub(".flv", ".html")
        link_list = File.open("#{output_folder}List of Links.txt", "a")
        link_list.write($BASE + link + "\n")
        link_list.close

        puts "Files created successfully."
    end

    folder = validate_directory(ARGV[0])             
    flvs = Dir.glob("#{folder}*.flv") 
    File.delete("#{folder}List of Links.txt") if File.exists?("#{folder}List of Links.txt")
    flvs.each { |flv| output_html_wrapper(flv, folder) }

Being DRY is about keeping a piece of knowledge in one place, whether that's a file name, the way you clean up some input or anything similar. The normalize_directory method I added DRYs up your code by putting in one place the logic for making a valid directory name out of the user's input.

Declaring a variable for a single use is extremely useful if it helps describe what the value is being used for. The vid variable in your code is still a mystery to me regarding what it actually contains. This is a great place to give something a name, and in doing so document your code without comments.