Application Control uses a database of signatures. Once that database is downloaded to the gateway it can continue to be used even if the Internet is not reachable. The only real effect is that the APCL blade won't have the latest updates and will throw an "Attention" status, and may classify some traffic as "unknown" instead of some brand-new (or recently updated) application.
On the other hand, Check Point URLF does not use a fixed categorization database that is downloaded to the gateway (this is a common misconception as many older products like Websense and SurfControl work this way), all categorizations are cloud-based via the rad process running on the gateway. If a requested site is not listed in the urlf_cache_tbl table a lookup is sent to the cloud by rad, as categorizations are received they are stored in the urlf_cache_tbl table for up to one week. When the response is received the user is allowed/denied access and the categorization result is added to the table.
If you have the URL categorization handling set to "Hold" the user's connection is not allowed to proceed to an uncategorized site until a response is received from the cloud. With this setting if there are Internet connectivity problems users will see delays or timeouts only when trying to connect to sites that are not already in the cache; the failures will look sort of random. You can set the URLF handling to "Background" and the users will be let go before a categorization is received, but this may allow users to briefly visit sites that they shouldn't prior to the cloud categorization being returned.
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