To correctly distinguish between administrative approval and technical sanction in civil engineering estimation, let's break down both concepts:

• Administrative Approval: This is the preliminary step in the process of implementing a project. Administrative approval is given by the competent authority of the government or organization that is responsible for considering the necessity and public benefit of a proposed project. This approval primarily focuses on the justification and necessity of the project.
• Technical Sanction: This follows administrative approval. Technical sanction is given when the designs, estimates, and plans of a project have been thoroughly examined and approved for technical correctness and feasibility. It ensures that the proposed works have been detailed out in the most economical way.

Now, let's evaluate the provided options:

1. Administrative approval checks material quality, while technical sanction determines project duration - This is incorrect. Administrative approval is related to project necessity. Checking material quality and determining project duration are parts of technical evaluations, but they are not directly linked to technical sanction as described.
2. Administrative approval authorizes procurement, while technical sanction approves staffing requirements - This is incorrect. Administrative approval concerns project necessity rather than procurement, and technical sanction does not specifically approve staffing requirements.
3. Administrative approval controls work execution, while technical sanction manages contractor payments - This is incorrect. Administrative approval is about recognizing project necessity, not execution control, and technical sanction is not directly related to managing contractor payments.
4. Administrative approval accepts project necessity, while technical sanction verifies design and estimates - This is correct. As explained, administrative approval relates to the acceptance of the project's necessity, while technical sanction involves verifying the technical accuracy of designs and estimates.

Based on the descriptions and evaluations above, the correct answer is:

Administrative approval accepts project necessity, while technical sanction verifies design and estimates