No they both have similar protections, but neither are truly sovereign territories. The 1961 Vienna Convention sets out rules governing consulates and embassies, and guarantees the “inviolability” of diplomatic premises. What this means is the host state can’t barge in without permission but it doesn’t mean the things that happen inside aren’t subject to the host state’s laws. The rule that allows consulates and embassies to act with their own laws is essentially a courtesy in some ways.

For a more historic example, the saudis who tortured and murder Khashoggi could have an international arrest warrant issued against them. The saudis wouldn’t hand them over but they wouldn’t be able to go anywhere else.

Another example of it not being sovereign territory is a baby born in a U.S. embassy does not have U.S. citizenship.

But they DO still have special protections and rules — for example an attack on an embassy is considered an attack on the country it belongs to.