Bash script to to split list based upon whether row item exists in secondary list?

An Awk solution:

Awk script file checklists.awk:

#! /usr/bin/awk -f

NR == FNR {
    # Recording first input script file (for you, IDList2)
    hash[$1] = $0
    next
}
$1 in hash {
    # if second file key is found in first file 
    print hash[$1]
    next
}
{
    # if second file key is not found in first file 
    print $1 > OUTPUT_FILENAME_NOT_FOUND
}

Change script execution modes:

chmod 755 checklists.awk

Awk script executed like that:

./checklists.awk -v OUTPUT_FILENAME_NOT_FOUND="./Output2"  IDlist2 IDlist1 > ./Output1

Warning, "IDlist2" file (the reference), is the first data file for Awk script.

Appears the standard join command meets your requirement.

Output1 (IDlist1 exists in IDlist2)

$ join idlist1 idlist2
TZ001
TZ004
TZ006
TZ007
$

Output2 (IDlist1 that does not match/exist in IDlist2)

$ join -v 1 idlist1 idlist2
TZ002
TZ009
TZ011
$