While the open architecture, open interfaces, and integration of intelligence within Open Radio Access Network technology hold the promise of transforming 5G and 6G networks, they also introduce cybersecurity vulnerabilities that hinder its widespread adoption. In this paper, we conduct a thorough system-level investigation of cyber threats, with a specific focus on machine learning (ML) intelligence components known as xApps within the O-RAN's near-real-time RAN Intelligent Controller (near-RT RIC) platform. Our study begins by developing a malicious xApp designed to execute adversarial attacks on two types of test data - spectrograms and key performance metrics (KPMs), stored in the RIC database within the near-RT RIC. To mitigate these threats, we utilize a distillation technique that involves training a teacher model at a high softmax temperature and transferring its knowledge to a student model trained at a lower softmax temperature, which is deployed as the robust ML model within xApp. We prototype an over-the-air LTE/5G O-RAN testbed to assess the impact of these attacks and the effectiveness of the distillation defense technique by leveraging an ML-based Interference Classification (InterClass) xApp as an example. We examine two versions of InterClass xApp under distinct scenarios, one based on Convolutional Neural Networks (CNNs) and another based on Deep Neural Networks (DNNs) using spectrograms and KPMs as input data respectively. Our findings reveal up to 100% and 96.3% degradation in the accuracy of both the CNN and DNN models respectively resulting in a significant decline in network performance under considered adversarial attacks. Under the strict latency constraints of the near-RT RIC closed control loop, our analysis shows that the distillation technique outperforms classical adversarial training by achieving an accuracy of up to 98.3% for mitigating such attacks.