What is the diference between ASA Transparent mode and routed Mode , what sevices are enable bydefault on tranparent mode and router mode?

Routed Mode is like forwarding done by firewall is based on L3 information whereas in transparent mode forwarding done by firewall is based on L2 information , still traffic is inspected in same way(L3 to L7)

By default firewall is in routed mode, if you looking for some particular services we can discuss first you have to mention them.

Like PAT is supported in routed mode but not in routed mode.