I am going to change my working sphere from Enterprise Web Applications written for concrete business process to Public Web Sites that will be accessible to all users around.
What is difference between this two spheres at the most top level? What specific characters I need to know about modern web sites development?
In an enterprise application, functionality and efficiency trump aesthetics every time. This is because you have a captive audience. The people who use your application are being paid to use it.
However, when opening an application up to the public, aesthetics becomes more important. There are always alternatives, and a given person will be more attracted to the application which looks better. Granted, functionality is still very important for repeat users, but you won't get people in the door if your application looks amateurish.
As Zack mentions, public users have a lot less tolerance for poor UI than enterprise customers do. That said, public users are more tolerant of incremental change; you can upgrade a live site as you feel like it (as long as it works, of course!!) without having to go through endless feature-request prioritization committees and user-training requirements.
I suspect one could write books about this.
I suppose the first difference is the user base. With an enterprise, you can, at least partly, ensure the users are doing what they are supposed to - and if not you know who they are and where they live. Further, they can be fired for abuse. On a public web site, you almost have to assume that some part of your user base is not there for a positive reason. So be paranoid - if they're not attacking you yet, just wait.
A second related point is that users will find ways to use (abuse?) your site you never thought of. Plan for the worst, hope for better.
Third, language, culture and usage varies across the world. A form, for example, with "zip code" that accepts just 5 digits may make sense in the US but is useless in the UK. And asking for a state and restricting it to two characters likewise makes no sense say in Italy where Italy IS the "state". This also applies to actual content - that joke you think is so very funny may be offensive in other countries. And never under estimate the ability of some folks to be offended at anything.
Fourth - get a good bunch of beta tester and test your site, and updates, carefully and thoroughly.
Fith, have a plan for scalability - if you suddenly get "discovered" can your site take the traffic.
That's 5 things at least.
Browser agnosticism - In enterprise apps, it used to be that the developer would target the app at a specific browser, just for simplicity's sake. In internet accessible apps, the developer must target the vast majority of browsers. While this has gotten easier in the last few years, it is still a issue that needs attention.
Scalability - its easier to scale an enterprise app, its easier to predict the growth of usage of the app, or simply design for access by all users in the org at once. This is not generally the case for internet sites. The day you get slashdotted, or dugg is the day that you learn this. Better to design scalability in from the start, rather than have to learn it at the time that your site starts to suffer.
In addition to Zack's answer, I would say that a web site/application that is open to the public needs to be constantly evolving/refreshed in order to grow your user base and keep them. Whereas on a more closed system, consistency and reliability are key priorities.
Depending on the nature of the application, if it has significant amounts of content Internationalization and presentation of content are hugely important.
Public web sites needs to be easy to use. While it's important that they look somewhat polished, don't ever let polish get in the way of ease of use. For example many designers like fixed width layouts because they are more predictable, many users like fluid width layouts because they use the space more efficiently. Side with your users.
Enterprise users can be forced to deal with needlessly-complex systems (lord knows I am more than I'd like), the general public cannot.