vCenter Server Appliance (vCSA) is the server that controls and manages your VMware environment. I would suggest using this over the vCenter Server Windows install. It has the advantage of easier install and manageability and seems to be the method most preferred by VMware, as well.
Much of the management of the VMware environment can now be done from the web interface, but you can still use the vSphere fat client, which is still my preferred method of interacting with the VMware environment. The web interface is better than in past iterations, but still just lacks that little something that would make me voluntarily to move to it on a permanent basis.
Improvements to My Organization:
VMware will assist your organization in managing computer and storage resources, and you will be able to shrink your physical environment significantly making it easier to manage in the long run. So don't expect immediate returns. There is a high price to be paid for licensing, hardware, configuration, and knowledge acquisition.
Room for Improvement:
VMware interacts with a lot of various hardware and has a vast array of capabilities and it seems that new capabilities are being added all the time. I would like to see more documentation and assistance provided by re-sellers to assist in the initial design of customer environments.
Cost and Licensing Advice:
There are 3 levels of licensing: Standard, Enterprise, and Enterprise Plus. VMware is very pricey and really the most useful in very large virtualization environments. It takes a lot of hardware and configuration knowledge. If you don't already have this in house, obtaining it can be pricey as well.
If you aren't already familiar with virtualization environments, I would suggest taking a course and/or doing a lot of reading before deciding on a license and configuration.
The best advice I could give is spend a lot of time designing your environment to receive VMware. Read everything you can get your hands on, and take a good online course or attend one in a city near you. Be aware that you will need to know a lot about computers, storage, networking, and security environments to determine the best design for you.
Download the latest ESXi version and build a test lab. You get 60 days to play with the full gambit of features. Then get a lot of advice from vendor specific engineers, HP, EMC, Netapp, Cisco, etc. because the hardware you choose will run into configuration issues specific to the vendors you choose to use in your environment. Don't go into this thinking you will see immediate returns on your investment. This is a long term design decision.