What do you like best?
Ease of use, dynamic quality adjustment over variable quality links, printer sharing, intuitive copy and pasting between machines, Simplicity of deployment in corporate environments. Clients available for every major platform and the mobile apps are now free, which is very useful for accessing devices out of the office via VPN.
What do you dislike?
The licensing model recently changed (and is much more complicated) so there's now no more perpetual licenses for server installs, which makes administration and software auditing more challenging.
When we purchased, the price break on per-unit licenses was nonsensical - about 10x as expensive per license, until you reached a high number of licenses (we obtained 1000 licenses at the price break point for the same cost as 100!) We had to effectively overbuy to get a good deal.
Recommendations to others considering the product
Depending on what you need, RealVNC may not be the best fit - it's the does-it-all Enterprise ready version produced by the original manufacturers. Installation's pretty simple, it's deployable if needed and it sports a lot of party tricks. However, TightVNC, UltraVNC or even TeamViewer may be what you need depending on your circumstances, requirements and budget.
What business problems are you solving with the product? What benefits have you realized?
VNC's very useful for quick, remote access of business critical machines where other access methods either cause software conflicts, interrupt operations or lock out the local user whilst you are remoted in.
Also using LDAP authentication, it's possible to grant or restrict access to subsets of machines in a granular fashion using our existing AD structures and role groups, plus it's one set less of static passwords to have to worry about changing. Great from an information security point of view.
We've reduced the amount of effort required for support departments to connect remotely and analyse problems, as they can see exactly what the user sees, in the same session, whilst they're on the phone.