In general you are encouraged to switch to the latest collection 6. Probably for most pixels the difference between the Collection 5 and Collection 6 data should be small, so empirical formulas developed for the Collection 5 are expected to keep working with the Collection 6 data.
While Earth Observing System data centers have switched to generating Collection 6 products, some direct broadcast centers are still running Collection 5 executables.
A comment on this post links to the Collection 6 MODIS Vegetation Index User's Guide. The Guide describes the differences introduced in Collection 6.
One of the differences deals with the pixels with high blue content such as snow, cloud or aerosol polluted pixels. Since MODIS blue channel saturates quickly the 3-band EVI algorithm was prone to errors. Collection 6 introduces a robust EVI backup algorithm that uses only NIR and Red data and therefore is not affected by the blue channel limitations. If your empirical formulas were trained on data that might be affected by the blue channel issues you might want to re-evaluate the formulas on Collection 6 data. Otherwise you should be safe using the new data since most pixels are unaffected.
Since your example tile covers areas with high level of near surface air pollution, it might be interesting to evaluate the differences in more detail. Such an assessment would make a complete answer to your question.