Publish = have (a book, periodical etc) printed and announce that it is for sale.

• He kept his writings with him and made no efforts to publish them.
• The University will soon publish the results  and the students can have them without further delay.
• The author found the cost of publishing by himself very high.  So he sold the patent to a publisher.

(2) make known to the public (limited usage).

• Publish the banns of marriage = announce formally in a church the names of persons shortly to be married.

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Publicise = give publicity to;  bring to the attention of the public.

• They’ve worked day and night to publicise their election campaign.
• He printed an excellent leaflet to publicise his pioneering work and its potentialities.
• Television and media have been under government control not to publicise controversies about the military and security matters.