How do independents choose a party to caucus with?
In the USA, you can be elected to the Senate as an "Independent". Notable examples include Bernie Sanders and Joseph Lieberman.
Being an Independent has a clear difference as far as the electoral process.
But once a Senator is elected as Independent, they usually declare that they will caucus with one or another political party (Democrats in both of the above examples).
Question is, what exactly is the practical political difference post-election between a Senator from a party and an Independent Senator caucusing with that party? (aside from official designation of "-D" vs "-I" in the suffix).
The question scope excludes the process of running for re-election.