Vulnerabilities in modern computers leak passwords and sensitive data.
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.
If your computer has a vulnerable processor and runs an unpatched operating system, it is not safe to work with sensitive information without the chance of leaking the information. This applies both to personal computers as well as cloud infrastructure. Luckily, there are software patches against Meltdown.
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
Spectre is harder to exploit than Meltdown, but it is also harder to mitigate.
However, it is possible to prevent specific known exploits based on Spectre through software patches.