Rome is the world's greatest museum city. Naples is the world's greatest street city. Rome has 2,800 years of layered history you study. Naples has 2,800 years of layered history you LIVE IN. Rome is elegant chaos. Naples is unapologetic chaos. Both are essential. The question is which one first — and the answer depends on what kind of traveler you are.
Food: Rome = refined traditions (cacio e pepe, carbonara, supplì, trattorias). Naples = raw intensity (pizza margherita at €4, fried pizza, sfogliatella, street food that makes Rome feel polished). Naples wins on food. Period. Safety: Rome is statistically Europe's safest capital. Naples has a reputation for pickpocketing and bag-snatching — the reality: the tourist areas (centro storico, seafront, Vomero) are safe. The reputation is 30 years outdated. Use common sense in both.
Cost: Naples is 30-40% cheaper. A meal in Rome: €15-25. Same quality in Naples: €8-15. Hotels in Rome: €120-200. Naples: €70-130. Museum tickets are comparable. Art: Rome dominates (Vatican, Borghese, Forum, Colosseum = unbeatable concentration). But Naples has MANN (the world's greatest collection of Roman art), Cristo Velato, Capodimonte, and Herculaneum/Pompeii nearby. My recommendation: First-timers → Rome first (easier, more iconic). Experienced travelers → Naples (deeper, rawer, more rewarding). Best: Both. 4 days Rome + 3 days Naples = the complete southern Italy experience.