One difference is simply this:
One can be a sinner who is godly.
One can be a sinner who is ungodly.
An atheist, for example, would be classified as 'ungodly' (without God), and an atheist is also a sinner (from God's point of view, which is that "death passed upon all men, for that all have sinned" - Romans 5:12.) Every person who dies has sinned, death being the 'wage' received for their sins (Romans 6:23). This includes atheists.
However, the context of 1st Peter chapter 4 indicates Peter to be thinking of (for example) murderers and thieves, some of whom could also appear to be people that believed in God and not be atheists. The term 'sinners' could apply to outwardly godly evil-doers.
But when Peter even includes those who are busy-bodies in other people's affairs, he could be thinking of Christians. He had just said that Christians should only suffer for bearing the name of Christ, and not for being murderers, thieves, evil-doers or busy-bodies. It all serves to show that even godly people sin. Godly people are less likely to be murderers or thieves than ungodly people, one would think, yet the warning about sin extends to even being busy-bodies! Has there ever been a person who has never meddled in other peoples' affairs? Unlikely!
Yet the term "ungodly and sinners" is brought into sharp relief in verse 17. This is where Peter's thinking is clear, and it is not merely about whether a person is openly ungodly, or apparently godly. Here is what he means:
"For the time is come that judgment must begin at the house of God:
and if it first begin at us [Christians], what shall the end be of
them that obey not the gospel of God?"
Disbelief in, and disobedience to the gospel, constitutes a person both as being ungodly, and a sinner, in Peter's writing.
So, given the wide range of sin warned against, it can be seen that nobody can wriggle off the hook of sin. Godly people sin. Ungodly people sin. Stating both categories of persons covers absolutely everyone, but predominantly, those who do not obey the gospel of God are both "ungodly and sinners".