A basic corded analog phone can keep working in a power outage, as long as the carrier's line stays powered. VoIP depends on power at your premises. The phone, the router and the modem all run on electricity there, and the network connection has to stay up as well.
The FCC's guide warns that some VoIP services don't work during power outages and that the provider may not offer backup power. Battery backup keeps calls going only if it covers every powered device on the path, the phone included, and only for as long as its capacity and the load allow. Where the service is set up to forward calls to mobile phones, that gives a second fallback.
Emergency calls differ for a related reason. An analog line sits at one address, so a 911 call from it comes from a known place. A VoIP account can be used from more than one place, so the service needs another way to know where the caller is. The method varies with the service. One is an address held on file, which FCC rules call a Registered Location and 47 CFR 9.3 defines as the most recent information the provider has that identifies the user's physical location. The FCC's guide also says some VoIP providers have limitations on their 911 service. We cover both subjects in whether you can dial 911 from a VoIP phone and how to register a VoIP phone for 911.