CREATE TABLE AS with geometry in SpatiaLite

SpatiaLite requires you to explicitly register geometry columns in its ecosystem (updating metadata, adding triggers) using the provided management functions; SQLite does not support automatic detection like with PostgreSQLs type-modifier system.

Having created a new table as you specified you should be able to 'spatially enable' it with

SELECT RecoverGeometryColumn('points_berlin_3068', 'geometry', 3068, 'POINT', 'XY');