"The big lesson learned grom general relativity is that physics must be
background free."

The next sentence will either advertize LQG as a background free theory
taking this lesson serious or criticize string theory for apparently not
taking this lesson serious.

But a little reflection shows that the meaning of "background" in the
second sentence is not the same as in the first one.

GR is "background free" in the sense that an object in its action which is
not varied (integrated over in the path integral) in other field
theories like pure Yang-Mills for instance, now is varied
(integrated) - namely the metric tensor. Hence GR does not assume a fixed
metric background structure. (It still assumes a fixed topological
manifold, though for instance, which is not "varied".)

But if you look at the effective target space action of string theory
you'll note that it is precisely of the same "background free" form as
the Einstein-Hilbert action (no wonder, because it is just
Einstein-Hilbert plus dilaton/axion and higher order terms) and that the
metric indeed is dynamical, just as in GR. That's pretty obvious.

But now when a theory is quantized there is a new notion of "background"
arising. Quantization means evaluating a path integral and perturbatively
quantizing means approximating the path integral at a saddle point and
then computing corrections to that approximation.

So when we say "quantizing theory X on background Y" we mean "Pick the
saddle point of the path integral given by the classical solution Y of
theory X and then compute quantum fluctuations of it".

This notion of background may seem similar but is completely different
from the one above. Every classical solution of a theory X which is
"background free" in the first sense is a "background" in the second
sense.

And this already tells you why "background freedom" in the first
sense is good (everything should be dynamical), while "background
freedom" in the second sense is bad - it would mean that there are no
classical solutions of your field theory!

String theory is nicely "background free" in the first sense, even more so
than ordinary GR, for instance. Not only is the metric a dynamical
quantity, but even the number of (macroscopic) dimensions, the coupling
constant, and to a large extend the entire field (particle) content of
the theory is not a fixed ingredient of the Lagrangian but is dynamical.

That's the very reason why one can even consider dynamics in the string
theory "landscape". String theory is so immensely "background free" in the
first sense of the word that it is at present very hard to say anything
about which values all these dynamical quantities it contains actually
will obtain after some evolution. The landscape discussion is one of
dealing with a theory which is highly background free, so that you first
have to solve equations of motion to even be able to say something about
the matter content of the theory. This is quite in contrast to other
approaches to quantum gravity, which are often considered to be truly
"background free", where all the matter content, the coupling constants,
the number of dimensions is fixed by hand.

So the problems with answering the "landscape question" in string theory
should be seen in light of the fact that they arise due to a difficult
question which other theories can't even ask.

This directly leads to the next point: String theory is certainly not
"background free" in the second sense. It does have classical solutions!
And that's good. These classical solutions can be used for perturbative
quantization by using them as a "background" about which to start a
perturbative expansion. But it must be realized that this notion of
"background" is one of how to do practical computations, not one of
principle.

In principle you could use other tools to compute the quantum theory, like
for instance computing in non-perturbatively. And there are ways to do
that in string theory, too, but these are as yet not completely general.

It is this non-perturbartive quantization which people really have in mind
when they say that, for instance, "LQG is a background free quantization
of gravity". This just means that, indeed, LQG is an attempt to quantize
gravity in one stroke, without perturbing about classical solutions of it
(="backgrounds" in the second sense).

But currently it seems that LQG, which is certainly "background free" in
the first sense of the word (the metric is dynamical) is maybe even
"background free" in the second sense of the word - which however
would mean that it admits no classical solutions! But that would be very
undesirable, since the world we perceive is so obviously well described by
classical gravity to good approximation, that it is a great challange for
experimentalists to find any quantum fluctuations (of gravity).

(Personally I feel that it is maybe not such a great surprise that the
quantization method used in LQG has problems finding sensible solutions,
since we know that when applied to 1+1 dimensional gravity it does not
reproduce the correct path integral quantization. In 1+1 dimensions there
is a well known theorem that when you solve all the diffeomorphism
constraints the Hamiltonian constraint cannot have any solutions at all.
This seems to be precisely the problem encountered for years in 3+1d LQG,
too.)

Next, what is the difference between "background freedom" and "background
independence". Well, I guess when people use "background freedom" in the
first sense (meaning that the metric is dynamical) it is pretty much
synonymous to "background independence" in this first sense.

But in the second sense of the word "background" (=classical solution
used in quantum perturbation theory) the  term  "background indepence" is
an important concept. It refers to the question if the results obtained by
perturbing about one background will coincide with the results obtained by
perturbing about another one. This is a nontrivial question and an
important consistency check of perturbative string theory. And indeed, it
can be seen in various nice ways that perturbative string theory, while
requiring the specification of some background (=classical solution) is
independent of this choice. So it is a fixed but arbitrary background (in
the second sense) that is used in perturbative string theory.

And that's a good thing.

P.S.
See the recent discussion on SCFTs and their relation to "backgrounds" to
see more fine print to that last paragraph.