"Communicating via computer" can mean many things, and the advantages/disadvantages are different for each.
In general, communication via computer provides people who are not able to be with each other the opportunity to stay in touch through the exchange of written words, live written conversation, live visual conversation, and many other options. For people who are separated from loved ones and cannot afford or are unable to travel, computer communication is a wonderful and useful tool in both personal and business settings. It is not, however, an equal substitute for face-to-face conversation in all circumstances.
When on-line communication is used instead of "real" communication, the speaker is not developing the social skills which are so necessary in the workplace as well as in successful relationships. Communication on line does not have to be based in truth and there is no certain way to know that, at least without doing some research; many people have been duped--or worse--by this kind of deceptive communication.
The most effective form of personal communication is two-way communication, in which both parties are able to express themselves and get feedback from the other party in as many ways as possible, including tone of voice, inflection, body language, among other things. Any computer communication which can do that is much more effective, and the same is true in face-to-face communication.
In the broadest terms, unless there are extenuating circumstances which make in-person communication impossible, communicating only over the computer is not ideal; however, it is a useful tool for businesses and individuals who cannot meet in person.
I have included a link, below, which claims that "in a few years, men will be able to communicate more effectively through a machine than face to face." That may be true, but it seems to me it will not be a better world.
To clarify the question first, there are a lot of ways of communicating via computer. The most common form is through writing and delivering this written communication through Internet (email, social media, other chats/forums/blogs/websites, etc.). By now, however, lots of people also communicate via computer through video, such as through Skype. There are also ways to communicate via computer only through audio (with the video of Skype).
Regardless of the form of communication, the biggest advantages of using computers to communicate (mostly through the Internet) are speed and cost-efficiency. It is much faster for most people to send an email rather than type a message, print it out, and send it via snail mail or deliver the message in person. Given the time difference and using the idea that time is money, this is also more cost-efficient. The farther the physical distance between sender and receiver, the more cost-efficient the computer option seems. It costs more than normal to make international calls, but if you already have Internet service for normal web browsing, using the Internet to transmit emails to people in other countries is much cheaper.
The biggest disadvantage of communication via computer is lack of personal contact and all the social cues/nonverbal communication that go along with that contact. This is most apparent in text communication, where the reader may not pick up on the tone of the writer. This can lead to miscommunication and misunderstandings. Another disadvantage of communicating via computer is the sheer volume of information we now have to deal with. This can show itself as an advantage (wow look at all our resources!), but having all this information available has led to decreased attention span of individuals reading webpages, and it can be harder to focus on dense information when that information is communicated via Internet.