Actually, the single quotes are making the difference, not the double ones:
Code:
awk -F'|' '$1~/'"$1"'/{ code... }'
All the awk code is enclosed in single quotes, when you want to "break" the awk code because you need the value held by an external shell environment variable, you simply close your code with a quote and reopen it (with another quote) when you've finished the external variable evaluation.
Putting double quotes around a shell variable is only a good scripting practice (so you avoid possible problem with null values, etc...).
