The thirteenth amendment did indeed end slavery; but it bears no relationship to the Emancipation Proclamation, nor was it a requirement for Southern states to return to the Union, as it was generally accepted that the Southern states had never left the union. It is more correct to state that, because they had engaged in rebellion, Congress would not seat the senators and representatives from those states until the amendments were ratified, including the 14th. It was necessary because slavery was constitutionally protected, and there was no other way to be rid of it. The Emancipation Proclamation was a wartime measure which had no effect whatsoever other than to give the North the moral high ground in the war.
The 15th Amendment guaranteed the right to vote to all citizens, and was specifically intended to guarantee black men the right to vote. Obviously, some Southern states went to great lengths to prevent the vote; but the intent of the amendment was clear.
13th amendment: ratified in 1865, Abolished Slavery
"neither slavery nor involuntary servitude, except as punishment for crime whereof the party shall have been duly convicted, shall exist within the United States, or any place subject to their jurisdiction."
Banning slavery was constitutionally necessary because of the Emancipation Proclamation made by Abraham Lincoln in 1863, where the Southern States had to agree to give up slavery in order to come back into the Union.
15th amendment: ratified 1870, Right of Suffrage(The Right to Vote)
"The right of citizens of the United States to vote shall not be denied or abridged by the United States or any State on account of race, color, or previous condition of servitude."
This amendment only allowed men to vote. States did find ways to prevent blacks from voting, such as with the Jim Crow Laws and the KKK. Many blacks died during this time period to excercise their right to vote.
Amendments 13 and 15 are part of the Reconstruction Amendments.
The Thirteenth Amendment, 1865 abolished slavery and involuntary servitude. It completed the task of the Emancipation Proclamation.
The Fifteenth Amendment, 1870 provided the right to vote for Black males.