I would suggest that the biggest problem solved is one experienced by merchants rather than private individuals. In particular the ability to receive electronic payments with much smaller transaction fees in comparison to the current common solutions (Visa, Mastercard, Discover, American Express, Paypal, etc). As well as protection from fraud perpetrated by individuals who would take advantage of the "charge-back" feature of most of the common solutions.
This is a big thing to me and all merchants. Paypal etc charge a percentage that is significantly greater than Bitcoin (and if it is international even more). Then as has been pointed out if they decide to do a chargeback the seller is the one who takes the loss. It is very difficult (nearly impossible) to get paypal to remove a chargeback without at least a 1-2 week back and forth with the customer and if you don't have proof of delivery (i.e. Brazil...if you send the package the tracking stops at the border and it will never show delivered...Paypal will not correct it without proof of delivery). I have had these issues in the past when I was an ebay seller and it is a large part of why I do not deal with ebay anymore.