Could there be a difference to the human body between Low Earth Orbit “zero G” microgravity and midway between planets weak gravitational fields?
tl;dr: day-to-day routine movement and scheduled exercise will dominate the microgravity experience of long-term missions; whether you orbit the Earth or the Sun, there won't be any meaningful difference.
I think these days we usually prefer to call the residual accelerations experienced by people (and things) in LEO as microgravity rather than the colloquial weightlessness or zero-gee:
Microgravity environment is more or less synonymous in its effects, with the recognition that g-forces are never exactly zero.
From a physics perspective, the only thing that matters is the sum of all sources of acceleration. In LEO there's still ~0.9 g gravity since we constantly accelerate in response we don't experience it.
Astronauts not touching the sides of their container are in free fall. Or at least their centers of mass are. There's a teeny tiny gravity gradient across the body, but I think that if they simply raise their arms to scratch their noses, they'll experience larger acceleration gradients in the process.
Most of the issues related to microgravity in an astronaut container in LEO involve the physical relationship between the astronaut and the container. If you start "at rest" with respect to the container at its center of mass (or along the orbital arc that passes through it, you'll stay there. However if you start a little "higher" or "lower" the orbital trajectories will differ and from your perspective inside the container you will think that you start accelerating towards the container wall.
If you grab something to stop, you'll now experience a slight acceleration and you'll feel a teeny tiny force in the process, and you'll say "Hey, microgravity!"
If the container performs one of its regular altitude raising propulsive maneuvers needed to counteract orbit lowering from atmospheric drag, astronauts then experience much larger microgravity, but usually for only several minutes, and there's going to be some warning.
LEO vs Helio
The main difference between being in a container in LEO vs a container on a heliocentric elliptical transfer orbit to Mars is degree.
Accelerations due to Earth in LEO and due to the Sun in Earth-Mars transit are about 8.7 and 0.003 m/s^2, so the primary microgravity effects will be roughly 2500 times smaller.
The tiny effects due to the gravity gradient across the astronaut's body will be reduced by a factor of a hundred million.
So what's the answer?
Astronauts in LEO experience tiny accelerations as the work and move about the container, pushing themselves off from the sides to get from point A to B, and from the straps they might wear while sleeping to help "stay in bed".
They receive major accelerating forces when they work out each day (exercise) to fight dangerous muscle and bone density loss.
Most of those "doses" of acceleration will almost certainly take place on a Mars transfer mission as well. Assuming they'll land and try to walk around, even though Mars' surface gravity is lower, they'll still need to be able to get out of their seats, stand up and walk around.
So I'm not seeing any real difference.