Darren's points are all essentially correct except for one thing. As far as I know, LiveCycle ES4 does not contain the full AEM 5.6 product. If contains a subset of it (CRX plus a few CQ bits required for Mobile Forms). In order to get full AEM functionality, you need to purchase AEM.
The LiveCycle product has been rolled into AEM 6 as "AEM Forms" and is available in two flavours: AEM Forms and AEM Forms for JEE.
AEM Forms is a subset of the LiveCycle modules (the ones that don't rely on JEE functionality) running under the AEM OSGi container. This version has most but not all LiveCycle functionality (in particular, the Process Management and orchestration engine are not part of this product).
AEM Forms for JEE is essentially the upgrade path from LiveCycle ES4 to AEM Forms. It is structured in a way that is similar to ES4 (it has both the traditional LiveCycle and AEM code running under a JEE Application Server). If you are familiar with LiveCycle, then AEM Forms for JEE will be very familiar to you. This version has full LiveCycle and full AEM capabilities.
If you know LiveCycle and you don't need the AEM functionality, then you can install AEM Forms for JEE and continue using just the LiveCycle components. Things won't feel much different than any other LiveCycle upgrade to you.
The upgrade from LiveCycle to AEM is not covered directly under LiveCycle maintenance and support however the sales representative you bought the software from should be able to negotiate a deal on upgrading to AEM.
Regards,
Rob