Approaches in demand analysis

What is the difference between Engel Curve and the system approach of demand analysis?

Take the setting of two goods, $$q_1$$ and $$q_2$$ with prices $$p_1$$ and $$p_2$$ and total income $$y$$. Then the demands can be written as: $$q_1 = d_1(p_1, p_2, y)\\ q_2 = d_2(p_1, p_2, y).$$
If you look at Engel curves, you consider prices as fixed, so the only thing that is allowed to vary are the income levels. In this case, you can simply write: $$q_1 = e_1(y),\\ q_2 = e_2(y).$$
If you want to estimate the full demand system (i.e. the functions $$d_1, d_2$$), you have to make sure that you have variation both in income levels ($$y$$) and (relative) prices ($$p_1$$ and $$p_2$$), so either time series data for a given individual or panel data for a group of individuals.
If you do not have price variation (e.g. because you are using a cross sectional dataset and everyone faces the same price, or if prices do not change over time) then you can only estimate the Engel curves $$e_1$$ and $$e_2$$.