This paper proposes an effective and novel multi-agent deep reinforcement learning (MADRL)-based method for solving the joint virtual network function (VNF) placement and routing (P&R), where multiple service requests with differentiated demands are delivered at the same time. The differentiated demands of the service requests are reflected by their delay- and cost-sensitive factors. We first construct a VNF P&R problem to jointly minimize a weighted sum of service delay and resource consumption cost, which is NP-complete. Then, the joint VNF P&R problem is decoupled into two iterative subtasks: placement subtask and routing subtask. Each subtask consists of multiple concurrent parallel sequential decision processes. By invoking the deep deterministic policy gradient method and multi-agent technique, an MADRL-P&R framework is designed to perform the two subtasks. The new <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">joint reward and internal rewards</i> mechanism is proposed to match the goals and constraints of the placement and routing subtasks. We also propose the parameter migration-based model-retraining method to deal with changing network topologies. Corroborated by experiments, the proposed MADRL-P&R framework is superior to its alternatives in terms of service cost and delay, and offers higher flexibility for personalized service demands. The parameter migration-based model-retraining method can efficiently accelerate convergence under moderate network topology changes.