Summary
My answer is yes, TG does generate sentences which cannot be generated by a CFG, but these additional sentences are not grammatical. There are two cases to consider, both involving extraction. (1) TG can generate sentences with extraction from one conjunct of a coordinate construction, while this is not possible to describe in a CFG. In fact, such sentences are ungrammatical. (This is described by Ross's Coordinate Structure Constraint, see Constraints on Variables in Syntax.)
(2) TG can generate sentences with more than one constituent, indeed an arbitrary number of constituents, extracted from a single constituent, while CFG cannot. But sentences with more than one extracted constituent from a single constituent are ungrammatical. (One instance of this, extraction from relative clauses, is described by Ross's Complex NP Constraint, Constraints on Variables in Syntax.)
If you have not actually read Ross's dissertation, you might reasonably guess from the title that the movement constraints he describes are now accounted for by the constraints he gives. Actually, they aren't, because Ross doesn't give any constraints on variables. The variables in question are those in the structural descriptions of movement transformations, standing for strings of labeled brackets and symbols. It is merely a conjecture that such constraints could be found and formulated, but they never have been (and now, I would say, we know why). So the Ross movement constraints are unsolved problems for TG.
I will append to this answer over the next few days an account of why CFG cannot describe extraction from just one conjunct of a coordinate construction and why it cannot describe extraction of an arbitrary number of constituents from a single constituent. This requires some knowledge of Gerald Gazdar's non-transformational theory of extraction.
Gazdar's theory of extraction
At first it had been thought by TG grammarians that the existence of unbounded dependencies created by extractions could not be described in a CFG. Around 1980, Gerald Gazdar found a way of describing them, after all, in a CFG. See, for example, Unbounded Dependencies and Coordinate Structure. Here is a rough outline of how Gazdar's method works.
To the non-terminal vocabulary of a CFG we append "slash categories", which are non-terminal symbols whose names have some ordinary category followed by a slash, followed by the ordinary category of a constituent which is missing. So, for instance, "S/PP" is the category of Ss which are missing a PP. A CFG must have a finite non-terminal vocabulary, but these new non-terminal symbols do not affect the status of the CFG, since there are at most the square of the number of ordinary non-terminals appended, and the square of a finite number is still finite.
We must also ensure that there are suitable phrase structure rules to interpret the slash categories. For each phrase structure rule with an ordinary category A on its left hand side (before the arrow), for each category B, there must also be a rule with A/B on the left, and on the right of the ps rule, (1) if B occurs on the right of the original rule, remove it, (2) otherwise for each non-terminal symbol C on the right of the original rule, replace C with C/B and require this ps rule to be in the CFG.
Sigh. I suppose that is not terribly clear. I'll come back to this and try to put it better. Intuitively, we add a finite number of ps rules to interpret the slash categories, and these additional ps rules either remove whatever is supposed to be missing, or pass down the requirement that it be missing from some daughter constituent.
An extraction is described in Gazdar's theory by giving a ps rule which has on its right hand side both the category of the extracted constituent and another slash category which describes the constituent that the extracted constituent came from. For instance, to describe a topicalized PP as being extracted from an S and Chomsky-adjoined at the left, requires a ps rule S -> PP S/PP. The construction is an S and has an extracted PP followed by a S which is missing a PP.
In adopting Gazdar's theory, we don't give up using movement to describe grammatical constructions, we only give up describing movement with transformations. Instead of transformations which transpose constituents over variables representing terms of a "proper analysis" that are incoherent strings of labeled brackets and symbols, movement is instead characterized in terms of tree structures, from a beginning state before the movement, to an end state after it, with the trajectory marked off by the slash categories of each constituent out of which the item moves.
Why only transformations can break Ross's CSC.
Coordinate structures have structures with at least 3 instances of the same category: each of the coordinated constituents has the same category, and the entire construction has that same category. The conjunction of Ss is an S, of two NPs is a NP, of two PPs is a PP, and so on.
When something is extracted from a coordinate structure, in Gazdar's theory, the structure has a slash category, meaning something is missing from it, and consequently each of the constituents of the conjoined structure has that same category, which will be a slash category. So it is not possible to extract anything from just one conjunct -- anything extracted must be extracted from all the conjuncts.
Why extracting many different constituents from a constituent is not possible in a CFG.
The slash categories describe constituents missing one constituent, but there is no way to describe constituents missing more than one constituent.
It will occur to you that although Gazdar's theory does not allow for the extraction of more than one thing from a constituent, it would be possible to construct CFGs which do extract 2, or even 3 things. It is possible. However, following Gazdar's method, it is not possible to extract any arbitrary number of things, because that would require an unbounded number of slash categories, A/B, A/B/B, A/B/B/B, ..., but the number of non-terminal symbols in a CFG must be fixed and finite.
