We have obtained deep spectroscopy of 26 planetary nebulae (PNe) and 9 compact HII regions in the nearby spiral galaxy NGC 300, and analyzed them together with the giant HII regions observed by Bresolin et al. (2009ApJ...700..309B). We have determined the physical properties of all those objects and their content in He, N, O, Ne, S and Ar in a consistent way. We find that compact HII regions have abundances similar to those of giant HII regions, while PNe have systematically larger N/O ratios and similar Ne/O and Ar/O ratios. We demonstrate that this nitrogen enhancement in PNe cannot be due to second dredge-up in the progenitor stars, since their initial masses are around 2-2.5M_{sun}_. An extra mixing process is required, perhaps be driven by rotation. Concerning the radial abundance distribution, PNe behave differently from HII regions: In the central parts, they show an average O/H smaller by 0.15dex. Their abundance dispersion at any galactocentric radius is significantly larger than for HII regions and many of them have O/H values higher than HII regions at the same galactocentric distance, suggesting that oxygen production in the PN progenitors in NGC 300 is common. PN abundance gradients in O/H, Ne/H and Ar/He are significantly shallower than those of HII regions. We argue that this indicates a steepening of the metallicity gradient in NGC 300 during the last Gr, rather than the effect of radial stellar motions.