Do different speaker stands can really make a big sound improvement?

Harbeth uses a lossy cabinet design, not a resonant one. If you think differently, I suggest you pull a woofer or drop the front panel and check out the bituminous damping pads attached to the cabinet walls. I assure you, they do not ring like a Steinway. That said, the recommended stands should be of the proper height, open design, and not couple the speaker box to the stand - use BluTak putty or small Sorbothane hemispheres at the far corners of the top plate to mount the speakers. Personally, I like the Pangaea 300 series with some autobody undercoat sprayed inside each tube to dampen them. They work well and are not egregiously priced. Mounting the speakers that way will allow the lossy, thin wall Harbeth cabinets to perform as intended. They have also worked well with my LS-50s high-mass cabinets at the 28" leg length. 

very important is the relationship of your ears to the tweeter

Did You Know?

Physics dictates that the stand supporting any loudspeaker establishes a mechanical relationship combining material science, geometry, and mass to functionality. This marriage impacts the sonic performance of the loudspeaker. The stand and speaker are linked to perform as a single vibrating entity. The same holds for floor-borne speaker systems and plinths or platforms.

The speaker stand is the difference maker. It determines how much or how little one hears and benefits from their loudspeaker. Locke Highleyman, ME Live-Vibe Audio

Do not forget stand must provide stability also.

A fallen speaker often results in dents, dings, and damage

I use Sound Anchors with my P3esr speakers.  The stands helped focus the sound and improved imaging.  I changed out the floor spikes with round headed carriage bold to avoid punching holes in my composite vinyl flooring (selected for acoustical damping;  the flooring is on solid concrete).  The speakers sit on small sorbothane pads on the corners.  With the carriage bolts installed I can drag out the speakers for a near-field experience when I want to, which increased involvement at the expense of a slight inconvenience.

I would recommend open-frame stands for all the other Harbeths, but the P3s work great on the solid stands.