What are the similarities and differences regarding the due process clause of the 4th and 15th Amendments?
I think you mean the 5th and 14th Amendments. The language is somewhat similar, as is the intent; the 14th rather clarifies the extent to which those protections extend. The 5th Amendment, part of the original Bill of Rights was written to protect citizens against abuses by the new central government; for that reason it seemingly applied only to violations by the Federal Government. The 14th Amendment, one of the Civil War Amendments, specifically provided that no state could deprive a person of life, liberty or property without due process. An examination of the language of the two illustrates the point:
The Fifth Amendment reads:
No person shall be held to answer for a capital, or otherwise infamous crime, unless on a presentment or indictment of a Grand Jury, except in cases arising in the land or naval forces, or in the Militia, when in actual service in time of War or public danger; nor shall any person be subject for the same offence to be twice put in jeopardy of life or limb; nor shall be compelled in any criminal case to be a witness against himself, nor be deprived of life, liberty, or property, without due process of law; nor shall private property be taken for public use, without just compensation.
The applicable language of the Fourteenth Amendment reads:
All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.
The purpose then, was to make the provisions of the 5th Amendment specifically applicable to the States. In fact, the Supreme Court has construed the equal protection clause of the Constitution to extend a substantial number of Bill of Rights protections against the individual states. Most defendants in court will allege violations of both amendments, as the intent is the same.