Wi-Fi systems based on the family of IEEE 802.11 standards that operate in unlicenced bands are the most popular wireless interfaces that use Listen Before Talk (LBT) methodology for channel access. Distinctive feature of majority of LBT-based systems is that the transmitters use preambles that precede the data to allow the receivers to acquire initial signal detection and synchronization. The first digital processing step at the receiver applied over the incoming discrete-time complex-baseband samples after analog-to-digital conversion is the packet detection step, i.e., the detection of the initial samples of each of the frames arriving within the incoming stream. Since the preambles usually contain repetitions of training symbols with good correlation properties, conventional digital receivers apply correlation-based methods for packet detection. Following the recent interest in data-based deep learning (DL) methods for physical layer signal processing, in this paper, we challenge the conventional methods with DL-based approach for Wi-Fi packet detection. Using one-dimensional Convolutional Neural Networks (1D-CNN), we present a detailed complexity vs performance analysis and comparison between conventional and DL-based Wi-Fi packet detection approaches.