I find the difference between the two to be difficult to talk about, since
they are closely related but never-the-less not isomorphic concepts, but
here is a shot.
In the standard formulation of Mechanics we say that there are 3 fundamental
dimensions: length, mass and time; and all mechanical quantities can be
expressed as combinations of these dimensions.
Hence the comment that energy has dimensions M*L^2/T^2 which is the same
combination of dimensions as torque. The two can be expressed with the same
units; both being Kg*meter^2/second^2 in the SI system of units or they may
be expressed with different SI units; the Joule and the Newton-meter. But
they have the same dimensionality. Note: that saying they have the same
dimensionality does not mean they are the same physical concept! This is
the main reason for using derived units like the Joule or Newton-meter, in
order to have some different units for different physical concepts.
Here is my favorite example to illustrate the difference. When we meaure
angles in radians. The units are Radians, to distinguish measuring angles
in other units such as degrees or grads. However, the angle measusred in
Radians is dimensionless as its definition comes from taking a length and
dividing by a length. So here we have a quantity that dimensionless but not
unitless. (Note: I have seen some people say that the Radian is unitless;
but I consider that to be a statement made by folks that have midunderstood
the difference between units and dimensions.)
Side Note: The statement that the standard formulation has three fundamental
dimensions is an arbitrary decision. You can develope a consistant set of
units for mechanics based on two fundamental dimensions, say length and
mass. This is in effect what is being done by the people who set c=1 in
special relativity.