To respond directly regarding the Materials Project data, I'm a staff member there so maybe I can shed some light.
Materials Project computed data is currently generated using a technique known as Density Functional Theory (DFT) with the PBE exchange-correlation functional. This results in some well-understood, systematic differences from experiment. Typically, this means that our computed lattice parameters will over-estimate experimental lattice parameters by 2-3% on average. Note that layered materials (any material where van der Waals bonding might be significant) will have larger errors in their inter-layer distance. Finally, note that these lattice parameters are nominally at 0 K, and do not take thermal expansion into account.
Note that lattice parameters on Materials Project are often given as their primitive cell, if you want the conventional lattice parameters make sure to download the CIF file in the "conventional" setting.
Band gaps will be systematically under-estimated by a large degree when using PBE (see our documentation). Spin-orbit coupling is also not included. The electronic band structures on Materials Project are most useful for seeing the shapes of the bands and the character of the gap (e.g. indirect, direct, between what symmetry points, etc.), the absolute magnitude of the band gaps are only useful for trends between different materials.
Better computational techniques can give results with smaller systematic errors, and we're constantly evaluating using some of these better techniques with the Materials Project. The trade-off here is that Materials Project tries to calculate properties for 100,000s of materials, and so using these better techniques is not always practically possible due to their computational cost.
With this context, to answer the question of "which should I use?", the question depends on what you want to use it for. If you want to know the "true" value, always defer to high-quality X-ray diffraction (bearing in mind the experimental value might be affected by grown-in strain, impurities, the temperature the measurement is taken at and other factors). However, if you want to do additional calculations with PBE, it's often easier to start from the previously-computed geometry. The computed geometry is also useful for examining differences between materials (e.g. varying composition) and also for materials where high-quality experimental data has not been acquired.
Likewise, for band gap, I would always defer to the experimental value, but of course there are also experimental issues too; experimentally, the optical gap is usually what is measured (e.g. via photoluminescence), there might be defect levels, finite temperature effects, excitonic effects, unintentional doping, Moss-Burstein shifts, etc., you might be only measuring the direct gap, whereas computationally you're predicting the fundamental gap (strictly speaking, the "Kohn Sham gap" using traditional DFT, which is another very important but subtle point). So there's no easy answer for which is better. The computational picture might give you a better picture for how a hypothetical pristine material might behave, but is typically most useful for trends and comparisons between similar materials.
Hope this helps! Happy to answer further questions.