Yes, they CAN be quite different. One factor is rigidity. You might want a stiff one for walking, but a gel mold one for running. Also where they correct your pronation (in the rearfoot with a wedge at the heel, or at the forefoot with a varus wedge, a wedge behind the ball of the foot at the front of the arch, or with more arch support and more stiffness at the rear arch) will do different things in each activity. A heel wedge won't help your leg alignment on the bike, for example, because you are connected to the pedals with your forefoot. Treating the forefoot for running (a heel-strike sport) may not be sufficient to correct your gait if you need a varus/valgus wedge at the heel.

I think that if you mostly need to have the arch bolstered up that the orthotics will be the most adaptable to all activities, but it might not be perfect. I have custom orthotics for running with a higher arch and a wedge just behind the first metatarsal joint. Varus or valgus wedges in the forefoot (under the metatarsals) hurt, but by lifting just at the front of the arch, I can get aligned properly without traditional wedges. These orthotics work pretty well in my cycling shoes except for fit. (I should note that for knee alignment, I use Specialized shoes which have a varus forefoot wedge built into the sole). Because of that wedge, I use the Specialized high arch inserts. These give me enough support, and their "metatarsal button" helps my feet *almost* as well as my orthotics. My right foot needs a bigger wedge, and so with the Speicalized shoes, my toes still go numb more on that foot. But I make do, because I don't have anyone to make me custom cycling inserts.

If your tendinitis problems are linked to running but you have no problems cycling, then I would go to someone who can make you orthotics for running. Then if you can afford it, get some for cycling, but you may not need them depending on what they have to correct and what shoe you use. You might be able to just tweak your cycling fit by moving your cleats or getting more or less arch support from a basic cycling insert.