What is the difference exactly in modern smartphones?
In Computer Science, memory is a wide concept for everything that is capable of storing data, while RAM only stands for Random Access Memory, or more precisely, runtime memory.
Memory is the same word that the M stands for in both acronyms ROM and RAM. It's simply because both of them can store data, despite one being non-volatile and the other being volatile.
These concepts haven't changed even if you're talking about modern smartphones.
Fore more information, see Are there guidelines on how much free RAM a phone should have? and our tag wiki for
What is (free) RAM and when is (free) RAM important?
Whenever a process runs, it requires RAM to store is runtime data. When many processes run together, they demand more RAM. You should never make your phone's RAM full, so a portion of its RAM is available for use when a process requests more. That "available for use" RAM is called free RAM (or more precisely, "available RAM").
Free RAM is always important, but it is often more important when you're trying to start another app or running a resource-intensive app. It is because when there isn't enough RAM, some processes may fail, and resource-intensive apps will refuse to run.
How can I as a user help it?
If you're using a moderately new phone, leave it alone because Android can handle it very well. Stay away from "RAM cleaner"s and "Task Killer"s as they do no help actually.
If you're using an old phone with 256 MB or so of RAM running Android 2.x, you should install less apps and routinely terminate them,
or throw it away and buy a new phone.