Shelley, assuming your road bike and spinning bike have the same bar height and extension I'm still going with the idea that you are becoming accustomed to a new activity. Just because it's a bike doesn't mean the technique is the same. I assume that your spinning bike has a fixed handelbar and that you can move your upper body in nearly any configuration without falling off? In other words it's a stationary bicycle and while your arms are necessary for anchoring your pelvis there's nothing needed in your upper body involving actual control of the bicycle.

If that's the case I'm guessing you are overcompensating with those muscles for control of the bikes direction or they are not needed to spin. Try this, next time you're out riding imagine using your latissimus dorsi muscles for posture and occasional power. In the pedal stroke on the same side those lats engage momentarily to anchor your hips to the same side of the bars, side to side, on each stroke. It's a momentary contraction correllating with the downward effort on the pedal. Without it people bob around.
Play with that muscle, in that easy spinning effort play with slight changes how the lats can engage as you change the arch of your back. Play with how the lats engage as you change effort in accelerating or climbing hills while seated.

I was never much of a racer but I discovered that was the first thing I had to do to begin to get power out of my whole body for time trial efforts and in the saddle sprinting. Those wings need to be brought into play.

This could be totally unrelated but I remember when I started racing a fellow at a bike shop was starting as well. He mentioned that he was getting head aches too often and one day he discovered he was scrunching his brow too much, he stopped doing that and the headaches went away. This is a round about way of suggesting that there is a feedback mechanism you have that you can discover. It may be that your posture on the bars in spinning class aren't appropriate for road riding all the time, it may be that you can allow your stomach to relax more while riding. Who knows but there's a light bulb waiting to be turned on. Personally I don't think it's equipment related if you're bars/seat setup is the same. Spinning on a stationary bike and riding on the road aren't the same activity.