Both Apostille and attestation is a method of legalization of documents for use in overseas. The procedure of attestation and Apostille  is different.

Apostille: Apostille is actually a square shaped sticker placed behind any document which tells the authenticity of document in abroad. Apostille is done for the country who sign The Hague Apostille convention of 1961. There are 115 countries who have signed the Apostille convention of 1961. The document legalization done by a member country is valid in remaining member countries.

Attestation: Attestation is a process of document legalization done for the countries  which are not a part of The Hague Convention.  You have to attest your documents from the embassy of the concerned countries to which you want to go. Attestation is a complex procedure as compared to Apostille.