The main difference is that with tinyproxy you whitelist sites by hostname/domain name, whereas qubes firewall whitelists by IP address (you can enter a hostname, but it is converted to an IP address when the configuration is saved). This means that if the hostname can resolve to multiple IP addresses (which is the case for websites behind CDN’s or load balancers) then tinyproxy will work, whereas the qubes firewall may work, but may also fail intermittently.
Depending on the type of site you are whitelisting, it may not be that hard to work out the list of required domains. The tinyproxy log file will list every site which it blocks access to, so you can work out which ones need to be whitelisted.
Also, the tinyproxy whitelisting uses regular expressions, so you can easily whitelist entire domains.