When comparing equally thick birch plywood and edge glued birch wood panels, what should I take into consideration in terms of their different properties?
The major consideration is wood movement.
A plywood panel is dimensionally stable and will respond minimally to changes in relative humidity, so fixings can be rigid and don't have to take any movement into account. With a solid-wood panel on the other hand movement — expansion and contraction across the grain only — must be borne in mind, especially on wider panels where it can be considerable.
Any decent furniture plans will naturally account for this in the design. Where it's a potential complication is with anything originally intended to be made from plywood but solid wood or a glued-up panel is substituted for the ply.
Other factors relate to strength and stiffness. Outright strength shouldn't be a big issue, few designs push materials beyond reasonable limits and in terms of how they're put together most are over-engineered so there's a good margin of safety. Stiffness on the other hand is something that often needs to be considered, solid wood being generally much stiffer than plywood, and the stronger the wood the more pronounced the difference (a brief look at the values in the Sagulator will give some idea). But where the material is used edge-up (not flat and spanning a gap, as in a shelf) this is far less of a concern.
While some like the look of plywood, I was considering some options closer to solid wood.
Just in case it's the visible plies and glue lines along the edges of plywood that's a major aesthetic issue for you it is common to edge plywood with solid wood where seeing the plies would be undesirable. This is very commonly done with shelves for example.
There's a little on this in this previous Answer.