What do you like best?
High reliability in terms of availability and uptime, which is essential.
It is easy to use, both standalone and to integrate with other applications.
Multiple different security options and configurations make it easy to create areas and files which are publicly accessible, as well as ones which can only be accessed either by specific people / groups, or for specific windows of time. This makes it idea both for general storage, and also for the sharing of more private information.
My greatest use of S3 is for podcast storage/serving, and for integration with my Wordpress-based sites.
It works very well in both cases - you can make use of their feature - rich API, or of the plethora of plugins available.
What do you dislike?
Generating time-sensitive URLs is harder than it ought to be, although this is pretty quickly overcome using a plugin.
Recommendations to others considering the product
Remember that S3 data are not owned by you, so make sure you keep your own local copy of all data just in case of any issues or data loss!
Take a little time upfront to map out your intended data structures, as it is harder to rearrange later.
What business problems are you solving with the product? What benefits have you realized?
Using it for several different things -
Backup storage space for off-site backup of certain files.
Speeding up websites by storing and serving files from S3 rather than own server, makes things much faster (especially when used in conjunction with a CDN);
Serving podcast recordings;
Sharing files with clients;
Distributing digital products once purchased