Distance is an issue. Often the cruise port is in a location far from the air travel gateway city. Good examples are Civitavecchia (for Rome), Le Havre (for Paris), and Seward (for Anchorage). In these cases, and others, the transit time from airport to cruise port can be literally hours. It is highly unlikely that you will be able to find economical private transportation relative to the cost of purchased transfers, unless you avail yourself of mass transportation (buses, trains or, ferries). Under those circumstances you are vulnerable to the vagaries of local transit, from mechanical problems to less than slavish attention to punctuality. (As anyone having experience with the Italian railroads can attest).
In addition, cruise lines often combine long-distance transfers with features of shore excursions, so the trip is as much a guided tour as a transfer. This is often done in Rome, where the transfer to Civitavecchia includes a short tour of "the Eternal City," or between Costa Rica's capital, San Jose, and Puerto Caldera, which includes an overview of that country, its politics, economy and ecology. However, conversely, cities like San Juan, Miami or New York not only have short distances between air and sea ports, but plentiful and affordable taxis, especially if you team up with another couple or two, which can often make the taxi ride cheaper than purchased transfers.
Time is an issue. Think of time as the "Goldilocks Factor." If you want to go to the port independently, the amount of time between deplaning and embarkation can't be too short or too long; it has to be juuuust right. The issue with very little time between plane and ship is self-evident, and using cruise line transfers immunizes you against all but the most catastrophic flight delays or traffic bottlenecks.
The flip side of the coin -- too much time -- seems counter-intuitive, but can also pose a problem if you arrive in your departure port city early in the morning with all your baggage in hand, and the ship doesn't begin embarkation until that afternoon. This situation is not uncommon for European departures, with the lion's share of flights leaving the U. S. the night before, and arriving on the continent the following morning. In situations like this, the cruise lines often reserve public rooms in local hotels where passengers can relax, read, enjoy refreshments and socialize with fellow passengers, while the cruise line trucks their checked baggage directly to the port.
If language or logistics are issues. If the port you are departing from has a light volume of cruise ship departures, local taxi drivers may have difficulty navigating you to the gangway. In some cases the cruise port is in a corner of a much larger industrial facility, with poor or nonexistent signage guiding the way. (I once spent nearly an hour trying to get to a ship in Genoa because my driver kept going in circles trying to find a very well-hidden port entrance). In other cases taxi drivers in lightly cruised ports may not be up-to-date on security and documentation requirements to legally enter the port. Lastly, the percentage of service personnel fluent in English is directly proportional to the volume of English-speaking tourists that city has hosted. If American tourism is a recent development -- as in some cities in Eastern Europe -- language, for us Americans, can be a sticky issue. In all three of these cases, buying transfers may do wonders for your comfort level.