The three sets of Factset market cap fields are similar but one would typically want to use mkt_val, mkt_val_af, mkt_val_cf, mkt_val_qf, mkt_val_saf. Those generally align with the Morningstar mkt_cap field and is what one typically thinks of as 'Market Cap'.

The fields mkt_val_public, mkt_val_public_af are a subset of the above and represent the market cap minus the value of closely held shares. The 'closely held shares' of a company are typically owned by a few major shareholders and do not actively trade. If one wants to use market cap to calculate liquidity or turnover then it may make sense to use this value. Also, the difference between this value and the mkt_val above is the amount of a company which is closely held which may be insightful.

The fields mkt_val_secs, mkt_val_secs_af, mkt_val_secs_qf, mkt_val_secs_saf supposedly represent the market value of a particular share class. Remember that a stock ticker represents the stock of a company's particular class of shares not the company itself. While it is most common for a company to have a single share class and therefore we think of a stock ticker as 'the company', there are many companies which have multiple share classes. Google is a good example. There are actually three share classes (only two which mortals like us can buy)

GOOG_L class-A shares, also known as common stock, which have the typical one-share-one-vote structure.
GOOG class-C shares, meaning that these shareholders have no voting rights.
Class-B, which are held by founders and insiders that grant 10 shares per vote. Class-B shares cannot be publicly traded.

This last set of market cap fields tries to capture the valuation of the specific share class. However, from my experience these fields are often nan and often don't seem to make sense. I'd recommend not using these.

Hope that helps.