Yes, the weather can affect the satellite link. Rain and snow can attenuate the link to a point where it is unusable. Certain frequency bands are more affected than others. Remember that you need clear line of sight the the satellite from your location as well. A colleague of mine uses a system provided by Bentley Walker, he doesn’t seem to have much trouble with it.
I’ve just been investigating this and the following quote seems to cover the technical bases;
Because the Internet signal must travel 22,000 miles to reach a satellite and back to Earth, there may be a delay or latency between requesting data and receiving a response. The average period of latency for satellite Internet is between 500 and 900 milliseconds–much slower than a dial-up connection. This is tolerable when one simply needs basic Internet activities such as reading and sending e-mails or web surfing. But it may be too slow for applications that require real-time access, such as video conferencing, Internet gaming, voice chats and VoIP telephone calls
Satellite Internet may be affected by weather disturbances. Rain will not block feeds or signals but may trigger an interference called the “rain fade,” which can cause much slower download and upload speeds and an erratic Internet connection
The costs seem high as well with initial setup costing £200-400 and ongoing costs being about 3 times that of regular ADSL/VDSL style connections