In the initial question, the OP should distinguish between an embassy and a consulate or the consular section of an embassy. The difference very roughly is that the embassy is responsible for foreign policy, including economic (trade) and certain cultural programs, while the consulate performs services for USA citizens (not only passports but also tax help, documents for adopted children and marriages performed abroad, limited help for people arrested overeseas, etc.) and also for foreigners wishing to enter the USA (primarily visas). It's somewhat a distinction between big picture furthering of USA interests versus services for individuals.

In some foreign capitols, the consular section is located away from the embassy itself, so be sure to find the correct address and hours. In principle, there should be a 24/7 emergency contact phone number for USA citizens. If you can register your presence in the country on line, why bother visiting the consulate in person? IMO this is primarily for those going to risky areas and for long term residents abroad, not for short term tourists and business travelers under ordinary circumstances, but it can't hurt except for the time required.