I read that public officials blocking people on Twitter goes against the 1st amendment. So, what is the difference?
To be precise, blocking based on viewpoint was held to go against the First Amendment. The ruling would not apply to blocking for other reasons such as off-topic content, harassment, spam, and so on.
President Trump blocked someone on Twitter. A court got involved and said that this violates the First Amendment. This has been upheld by other courts who say that this stands, even on personal accounts.
It doesn't apply to personal accounts, only to accounts that conduct official government business. And the reason is important.
Blocking someone on Twitter doesn't just stop them from seeing the announcements, it stops them from participating in the discussion about the announcements. If the President could block people for disagreeing with him, then those who agree with him could use his Tweets to build a following and grow their influence while those who disagree with him could not. A government official having this kind of "thumb on the scales" of public policy debate is what the courts found a problem with.
And why is this any different than if a civilian who has a large political platform blocks someone they disagree with for such a reason? Isn't it a violation of the First Amendment whether or not it is an elected official?
Civilians can try to influence public debate however they want. If the government tried to limit the ability of non-government actors to influence public debate, that would violate the First Amendment.
The First Amendment prevents the government from unduly influencing debates over things like public policy. Preventing private blocking would do this. Permitting official blocking based on viewpoint would do this. So the government may not prevent private blocking and may not practice official blocking based on viewpoint.