This paper presents a detailed comparative analysis of two data-driven probabilistic forecasting methods - a density-based Empirical Copula (EC) and a non-parametric Quantile Regression (QR) - in the context of load forecasting for day-ahead market applications. In modern power systems with increasing stochastic renewable production, Distribution System Operators (DSOs) face the critical task of accurately forecasting production and consumption to strategically place bids in the day-ahead market. Any deviation from their dispatch plan can lead to the costly activation of reserve power. For DSOs with a small number of customers, or those requiring more detailed forecasts, traditional statistical methods may be inadequate due to their reliance on parametric statistical distributions of forecasted variables. This paper evaluates two probabilistic load forecasting methods that effectively operate without relying on parametric distribution assumptions: the EC and the QR. The associated forecasting models are identified with suitable optimization strategies and their performance is evaluated using standard probabilistic metrics on a real-world electricity consumption dataset. The analysis focuses on day-ahead 24-hour forecasts, and compares the proposed methods against a persistence baseline method. The results demonstrate the superiority of both EC and QR over the baseline, with the EC excelling in forecast accuracy, while QR showing dominance in prediction intervals.