Would one be able to tell the difference before landing? Not likely in many scenarios.
Case 1: Bird hits windshield: blood & guts are immediately visible. Drone hits windshield, cracks are visible but not much else. The crew CAN immediately figure out, if they've ever see a birdstrike on the windshield, that this was something else. (Of course, it could just be a shattered outer pane -- those do fail sometimes, not necessarily due to any collision. Never had one go in flight, so I don't know if the sound of the pane shattering might be mistaken for a collision {or vice-versa}, or not.)
Case 2: Bird hits near the cockpit but NOT on a windshield. Sounds like a sharp crack, like a piece of gravel hitting your automobile windshield at highway speeds. If a drone did the same, it would probably sound pretty similar, and without the blood/guts to see (or not see), the ability to notice what's missing may not be there.
Case 3: Bird hits the engine, and it starts running poorly. Sometimes you will smell the guts cooking in the core of the engine (i.e. they get into the bleed air system & thus into the air conditioning), and that's a clear sign that your engine ate a bird. But the bird CAN miss the core and still cause problems with the fan blades. Most birds that miss the core are chopped up by the fan blades & the crew had no idea until after landing that anything happened, but a big enough bird can cause damage that way. So if the engine starts running rough, hot, and with some vibration, the absence of "cooked goose" odor isn't necessarily going to drive the conclusion that the collision had to be with something other-than-a-bird. One might suspect that a lot of damage with no smell seems unlikely to be a birdstrike, but there are cases where some engine component has failed & caused engine damage (ranging from barely noticeable up to catastrophic), and you can't really know much about causality for the engine doing what it's doing until you're on the ground.
Now, all of this assumes that the crew doesn't see the object that they hit. Which they very well might. Everything in that case is obviously an entirely different discussion.