In ODBC administrator you can enable tracing. Then compare trace file from both slow and fast machine. If there is "fast" open from that machine using ODBC administrator and "slow" from your app then try other ways to open such connection. Try use it from other tool such as QueryTool (free trial), or create simple script in Python with win32 extension. In Python (I recommend Active Python which has win32 included) you can open ODBC with:
import odbc
import time
t_start = time.time()
conn = odbc.odbc('db_alias/user/passwd')
t_stop = time.time()
print('open: %.3f [ms]' % (t_stop-t_start))
cursor = conn.cursor()
cursor.execute("SELECT FIRST 1 DBINFO('version','full') FROM systables;")
for row in cursor.fetchall():
print('[%s]' % (row[0]))
(note Informix specific version select)