Embassies, as defined in the 1961 Vienna Convention on Diplomatic Relations, enjoy certain legal protections for the purposes of protecting their premises and (some) staff from interference by their receiving countries.
I'm specifically referencing such protections as restrictions on representatives of the receiving country entering the premises, i.e. a police officer cannot enter an embassy without the permission of the embassy.
Do the same protections apply to the premises of consulates?
More or less, but this manual gives details on the "less" part, as interpreted by the US State Department. See The 1963 Vienna Consular treaty for the general agreement. Facilities derive immunity from their relationship to consular personnel and their duties. That means that in some cases, a consulate has no immunity.
Diplomatic agents (such as ambassadors) and their families enjoy total immunity. However, some staff only enjoy "official acts" immunity. Immunities only apply to foreign nationals of the sending nation, and not to citizens or permanent residents of the US. Quite often, nations have consular offices outside the capital manned by locals with an interest in the sending nation, and they may carry out business in their own home. The individuals do not have regular consular immunity, they have only official acts immunity, and the premise where they do their business is not unsearchable, unless the premise is used only for official consular business (thus, not their own homes).
Under the Vienna treaty, "consular premises" is a term of diplomatic art, referring to a subset of consulates, as
the buildings or parts of buildings and the land ancillary thereto, irrespective of ownership, used exclusively for the purposes of the consular post
An honorary consul may uses his home for conducting consular business, but it is not immune from search (it is a de facto consulate, but does not qualify as a "consular premise" in the sense defined by the convention). Since such a consul does not enjoy absolute immunity, his residence is not un-enterable, as would be the case for an ambassador or regular consul.
Yes, as Dale says, the same protections apply to the premises of consulates and embassies. Under the Vienna Conventions, local authorities need the permission of the head-of-mission to enter the premises of either. However, "premises" are defined differently for embassies and consulates: the "premises" of an embassy includes more than the "premises" of a consulate, so more of the embassy is protected than of the consulate.
What protection do the premises of embassies and consulates get?
Embassies are covered by the 1961 Vienna Convention on Diplomatic Relations. Article 22 of that Convention says:
"The premises of the mission shall be inviolable. The agents of the receiving State may not enter them, except with the consent of the head of the mission."
Consulates are covered by the 1963 Vienna Convention on Consular Relations. Article 31 of that Convention, on the Inviolability of the consular premises says:
"Consular premises shall be inviolable to the extent provided in this article. The authorities of the receiving State shall not enter that part of the consular premises which is used exclusively for the purpose of the work of the consular post except with the consent of the head of the consular post..."
What exactly is protected?
As mentioned above, the definition of "premises" given in the two conventions are not the same.
The premises of embassies include all land and buildings
"used for the purposes of the mission including the residence of the head of the mission."
The premises of consulates however, include all land and building
"used exclusively for the purposes of the consular post."
So, at the very least, the difference is that consular residences, even if they are within the consulate, are not covered. (From the report of the working group that drafted the consular rules, it appears they did not include residences because those residences were not general protected under earlier treaties or laws.)
All diplomatic missions have the same protection whether they are called embassies, consulates, or high commissions.
