I use both and I slightly prefer the solvent-free type.  The solvent based is an excellent adhesive but can get stuck to everything it touches (including me) and is difficult to remove.

Solvent-free is cleaner to use and no fumes too so safer in an enclosed space.

As for strength I would imagine solvent based to be stronger and I use the solvent based to point in lead flashings on external brickwork.  I'm not sure the solvent-free would weather too well in that application.

The solvent-free is softer too so easier to squeeze out of joints avoiding 'stand off'