Meltdown breaks the most fundamental isolation between user applications and the operating system. This attack allows a program to access the memory, and thus also the secrets, of other programs and the operating system.
Spectre breaks the isolation between different applications. It allows an attacker to trick error-free programs, which follow best practices, into leaking their secrets. In fact, the safety checks of said best practices actually increase the attack surface and may make applications more susceptible to Spectre
Which systems are affected by Meltdown?
Desktop, Laptop, and Cloud computers may be affected by Meltdown. More technically, every Intel processor which implements out-of-order execution is potentially affected, which is effectively every processor since 1995 (except Intel Itanium and Intel Atom before 2013).