"Indian Spicy" vs. "Thai Spicy"

Well, that depends on the individual Thai dish or Indian dish and how it was cooked, of course. But I understand what you're talking about. However, the difference in heat sensations is not due to the kind of pepper employed. It's all about fat, really.

Frequently Thai dishes are made with fresh peppers, and have a lot of acid and salt in them (from citrus, tamarind, and other flavors) but very little fat (comparatively). Because of this, many Thai dishes have an instant burst of intense hotness which goes away realtively quickly. The paragon of this is probably Thai salads, like larb or green mango salad, which are highly acidic and very very spicy.

On the other hand, most Anglo-Indian food (familiar to Americans and British) is in the form of "curries" which use a slow-cooked dairy base (butter, milk, and/or cheese), and are spiced with dried ground or whole chile peppers. As a result, when you first taste them the fat conceals the capsicum from your tongue, gradually revealing it as your saliva breaks it down. Hence the "slow burn". (I've tried to find a medical reference for this to link, but have not been able to yet).

Similarly, Thai coconut milk curries can build up heat slowly and that heat sticks with you -- because of the hot peppers cooked in the fat of the coconut milk.

Incidentally, there isn't one kind of chile pepper used by either culture. The Thai have dozens of varieties of hot pepper and Indians have hundreds (as well as a dozen different regional cuisines, a few of which are not spicy at all). In the USA, these tend to get narrowed down to a handful of different pepper varieties (and substitutions like jalapenos) because of limited availability. The spice you call "red pepper" could be any of a half-dozen different ground dried peppers of varying hotness.

I tend to agree largely with fuzzychef's answer, as a lot of Indian dishes are based on spices in oil (usually a mix of garlic, ginger, onion and the Indian five spices (according to my local contact) of cumin, mustard seed, red ground chilli, turmeric and ground coriander seeds. The oil will absorb the capsaicin and other spicy compounds and spread them quite evenly trhough the dish, meaning that the "heat" gets delivered quite evenly and the fat makes it stick in your mouth for a while (resulting in an afterburn effect).

If you make a curry (Indian style with dairy like yoghurt or Thai style with coconut milk, that effect gets even bigger.

When you keep the ingredients seperated like in the case of fresh peppers in a salad, you will notic e a more immediate but shorter-lasting effect.

One Indian dish where that is clear (to give a counter-example) is papad massala, a simple fried pancake with onion, tomato, salt, red chilli powder and coriander leafs, which will burn immediately, more Thai-style :)

It depends on the particular dish -- many hot curries use red chili powder. Some dishes, like coconut chutney, use jalepeno or similar fresh hot peppers. I don't think there's a distinctive/unique spice as there is with Thai food; it's more in the combination of spices.

In addition to the other answers, I've found that cumin gives that sort of slow, background burn while cayenne and Capsaicin give that sharper, (for me) painful heat.