For travel, an eSIM usually wins. It gives you local mobile data, often at lower cost than roaming, and a local number if the plan includes one. A number service does not provide any data.
For privacy, a separate number is usually simpler. You can give it to marketplaces, sign-ups and short-term contacts without touching your main line, and use it from any device. With a carrier eSIM, the line is tied to one phone and, in many countries, to your identity documents.
For verification codes, line type matters. Many apps check whether a number is mobile, landline or VoIP before sending a code, and some refuse VoIP. A carrier eSIM with a number is reported as mobile, but only works while the profile is active on your phone. Internet-based numbers are convenient but can be refused. That gap is exactly what non-VoIP numbers aim to fill.