The power density is really nowhere close to a battery. Supercaps make sense for things where you actually need really massive charge and/or discharge spikes, over very short times. Think railguns, or a camera flash that can fire multiple times without needing to recharge between shots (if it charged enough to begin with), or possibly a smoothing system for charging batteries from a very spiky power source (hypothetically, this could scale to things like harvesting lightning, though at present that's not at all practical). They aren't practical for long-term storage, either due to energy density or due to their tendency to lose power over time pretty quickly.
A sufficiently large battery bank will have no problem with the charge speed of a photovoltaic array (which is actually rather slow). A small bank might reach saturation voltage - where the batteries are still charging but can't charge any *faster* or they'll take damage from overvolting - fairly quickly if fed by a large array, but that's not the real problem with a small bank; the real problem is not having enough storage capacity.
Expense is considerable, especially if you go with the low-maintenance options like gel-cells. However, supercaps are, at this time, not something you can buy a huge bank of at any price (certainly not the hemp-based ones). If you could get a meaningful capacity of the graphene ones it would probably cost many times as much. Maybe the hemp ones will change that, but don't hold your breath.
Maintenance is much less than it sounds. Wet-cells (typical lead-acid batteries) need topping up with water periodically, and occasionally may need equalization charges; the first can be done by a reminder to go do so every month, and the latter doesn't even need to be that often. Pretty much every other aspect of maintenance should be handled by a good enclosure for the batteries and a good charge controller. The controller costs a bit but you want one of the good ones anyhow; they perform DC-DC voltage conversion to take the output of the solar cells (which can easily be at least 25% higher voltage than the batteries will charge at) and down-convert it, extracting some extra current in the process (some energy is lost in this process, but it's typically a 10%-20% net positive for the 12V gel-cells my family uses). Speaking of gel-cells, those will save you on maintenance (at a cost of more money up front and a more severe voltage sensitivity that limits charge rate a bit harder). Such batteries are basically install-and-forget, but you'd need to be tremendously lazy for them to be worthwhile for a home installation; they are typically for marine usage (as my parents do) where never needing to open the cells (to add water) is a significant plus.