In designing neuromorphic circuits and systems, developing compact and energy-efficient neuron and synapse circuits is essential for high-performance on-chip neural architectures. Toward that end, this work utilizes the advanced low-power and compact 7nm FinFET technology to design leaky integrateand-fire (LIF) neuron and spike-timing-dependent plasticity (STDP) circuits. In the proposed STDP circuit, only six FinFETs and three small capacitors (two 10fF and 20fF) have been utilized to realize STDP learning. Moreover, 12 transistors and two capacitors (20fF) have been employed for designing the LIF neuron circuit. The evaluation results demonstrate that besides 60% area saving, the proposed STDP circuit achieves 68% improvement in total average power consumption and 43% lower energy dissipation compared to previous works. The proposed LIF neuron circuit demonstrates 34% area saving, 46% power, and 40% energy saving compared to its counterparts. The neuron can also tune the firing frequency within 5MHz-330MHz using an external control voltage. These results emphasize the potential of the proposed neuron and STDP learning circuits for compact and energy-efficient neuromorphic computing systems.