Differences Between Two Different Create Index Commands
It boils down to looking what the default values are.  Lets break this down:

CREATE UNIQUE NONCLUSTERED INDEX [DEID_MAP_IDX1] ON [dbo].[DEID_MAP]


nonclustered is specified here.  The default (i.e. nothing specified) is nonclustered.  So unless clustered is specified it will default to nonclustered.  So that's the same in both scripts.

[dbo] is specified here explicitly.  As for the second un-specified CREATE INDEX then it all depends on what the current user's default schema is.  Only you can answer that at the moment, so that may or may not default to dbo.

WITH (
    PAD_INDEX  = OFF, 
    STATISTICS_NORECOMPUTE  = OFF, 
    IGNORE_DUP_KEY  = OFF, 
    ALLOW_ROW_LOCKS = ON, 
    ALLOW_PAGE_LOCKS = ON
) ON [PRIMARY]


PAD_INDEX: the default is OFF, so unspecified will be the same in the second script as it is in the first.

STATISTICS_NORECOMPUTE: the default is OFF, so the second script unspecified has the same value.

IGNORE_DUP_KEY: the default is OFF, so the second CREATE INDEX is identical with this parameter.

ALLOW_ROW_LOCKS: the default is ON, so the second CREATE script has the same behavior.

ALLOW_PAGE_LOCKS: the default is ON...the second script has identical behavior.

... ON [PRIMARY]: just like the default schema one, this all depends on what your default filegroup is.  If PRIMARY is the default filegroup, your second CREATE INDEX script will also create the index on PRIMARY.  If PRIMARY is not the default filegroup, then it will be a different filegroup, as an unspecified filegroup will go to the default filegroup.

All of this information and default values can be found on this BOL reference here.