Yes, you're right-- there are different goals. The goal of generative linguists is as you and others here put it-- to develop a theory of the human language faculty within the theory of mind. There's not as much emphasis on a complete description of individual languages (although data points for all generative subfields-- generative syntax (word order), generative morphology (word structure), generative phonology (sound systems), etc-- still come from specific languages-- you can see this in any issue of the MIT generative journal Linguistic Inquiry). The goal of many descriptive linguists is to analyze specific languages, but of course, non-generative linguists have general theories about human language as well.
On whether non-generative linguists accept Universal Grammar or not, there's a wide range. Some say there's now experimental evidence that "general learning mechanism" approaches to infant language learning are more powerful than Chomsky and other generativists originally thought, and that this weakens his "poverty of the stimulus argument" in favor of UG. Others agree with the generativists on poverty of the stimulus, but think the Chomskyan separation between the syntax and semantics components harms the theories. Others think generativist "introspection" data collection methods are deeply flawed, and this makes their formalisms suspect (although there are generativists who working with more rigorous data collection methods ). Others, yes, do think that generativists are far too weak in data collection, and that generative theories are far too formalistic-- S. Robert Ramsey's The Languages of China (1987) makes this argument on p. 132. To give examples of all these perspectives, The Language Log weblog (http://www.languagelog.com/) has a good mix of non-generativists and generativists, and some great popular writing on linguistics. (They also have many amusing rants over gross journalistic misinterpretation of psychological research, cough cough, David Brooks, Ross Douthat.)
I will have to add the Kenneally book to my to-read list. Many years ago I read a little about Savage-Rumbaugh's work in Pinker, but he was skeptical of her pre-1994 claims about Kanzi, so I should probably read something that is more up-to-date and argues more in favor of Savage-Rumbaugh. Thanks for the information on the Michael Albert - Alex Cockburn ZMag thing, also.