Is wasabi considered to be spicy or to be a spice?

In my book, this is pretty trivial. Wasabi is absolutely a spice - it's something with a very specific flavor, derived from a plant, that can be used in fairly small quantities to add flavor to something.

It's not spicy (spicy hot, piquant) in the normal sense, though. It doesn't contain capsaicin. It is hot in some sense: it contains allyl isothiocyanate, which we obviously have a very strong reaction to. This is the same compound that's in horseradish and hot mustard. The reaction is very different from capsaicin, though. Capsaicin causes you to register heat at a much lower temperature than you normally would, so you're actually feeling heat, like you would if you were burning your tongue. It's quite literally hot, as far as your body is concerned. It's also an oil, so it can't be washed away easily with liquid, and the burning tends to linger. Wasabi, on the other hand, doesn't produce an actual sensation of heat, you feel it mostly in your nasal passages, and can easily be washed away with liquid, so it tends to be a brief sensation. So sure, "hot" is a reasonable way to describe it, mostly because we don't have a word for the actual sensation, but it's definitely not the same thing as a hot pepper.

I'm sure you can find plenty of people who would say that these are just two different kinds of spicy hot (piquancy), but arguing over definitions isn't going to get us anywhere. The important thing is that there's a fundamental difference, and there's no way you could substitute one for the other. If you want to understand why I think this (and it's not just a personal definition), go to any recipe site, search for "spicy", and see how many things with wasabi/horseradish/mustard supplying "spiciness" you find.

Edit: To stave off further debate in the comments, let me just repeat: arguing over definitions is not useful. There are probably a lot of people who think "spicy hot" should include this, and a lot who don't. I wouldn't generally expect anyone to think of wasabi without context when you say spicy hot, but you're welcome to use the terms however you and the people you talk with understand them.

I think the issue is primarily linguistic, but there may also be a mismatch between your experience of Japanese food and the average Japanese experience of Japanese food.

Let's start with the experience itself. Wasabi is generally used in moderation in Japanese cuisine, and when real, fresh wasabi is used, instead of the mustard/western horseradish mix that's more common, it's more pungent than spicy. That's a fairly nuanced distinction, and you may find both Japanese and non-Japanese that would use the word "spicy" to describe what amounts to a nasal reaction, instead of the more direct tongue stimulation that say capsaicin, or glutamates trigger.?In Japanese, you might say piri or piri-tto to refer to an abrupt sensation of pungency that doesn't linger, like (real) wasabi offers, or tsuun to refer to the tingling sensation in a more visceral onomatopea. Karai is used to describe spicy foods (and, in some cases, to describe salty foods, typically soups, but let's ignore that for now).

In any event, wasabi isn't really used as heavily in everyday Japanese cooking as its popularity in the US would suggest. Additionally, the US has latched on to spicy tuna rolls and complex, multi-ingredient gooey "rolls" as representative of sushi, even though in Japan most makimono are minimalist creations involving little more than some cucumber, or gourd, and aren't even the reason you go out to a sushi restaurant. The multi-ingredient ones with say egg and pickled vegetables are still simpler in taste than what most Americans would get excited about.

For many Japanese, seeing the ridiculous amount of reconstituted wasabi served with their little plate of nigiri-sushi or the sriracha augmented rolls comes as a bit of a surprise when they visit the US. Our culinary preferences tend to be adventure-seeking, whereas Japanese tend to have more interest in sappari (refreshing) or assari (light/subtle) flavors and are more focused on texture contrasts than intense flavors.

To some degree, wasabi is a regional food (Shizuoka prefecture grows much of it), even though it's found around the country thanks to modern distribution. Sushi is not an everyday experience for most people, either, and it's not seen as a "spicy" thing when it is consumed, because most people don't eat it with loads of wasabi; they want to taste their fish.

From a culinary perspective, mustard is one of few "spices" that wouldn't actually be referred to as an herb that's really used in Japanese cuisine. (It's also a major component in mass market wasabi). Ginger is an exception, though it's also mostly used sparingly, and generally fresh, so it is only arguably "spice".

The "spicy" flavors that are popular in Japan are probably the Japanese interpretation of English-style stews called "curry". These use Indian blends of spices adapted to Japanese tastes, but most versions are sweeter and milder than they are "hot". It's somewhat common, but not necessary, for people to enjoy extra spicy curries. But curries have a status that is vaguely foreign, like tikka masala or mulligatawny soup in England, even if both are really "local" innovations. Even if you're Japanese, you may not quite consider curry as a spicy "Japanese" food.

Additionally, you may notice even in English, the notion of "spice" isn't perfectly attached to the notion of "spicy." If I use cloves or ginger in something, it might be "spiced" with spices, but perhaps isn't considered spicy.

Spice is defined in the Chambers 21st Century English Dictionary as:

spice noun 1 any of various aromatic or pungent substances, such as pepper, ginger, nutmeg, cloves, cinnamon, etc that are derived from plants and used for flavouring food, eg in sauces, curries, etc, and for drinks such as punch.

While wasabi is sometimes added directly to food during cooking these days, traditionally it is made into a paste and served as an optional side dish - a condiment, as Mischa says.

However, while it is not truly a spice, that's not to say it is not spicy, which has the colloquial meaning of not only tasting of spices, but also tasting hot, like, say, black pepper or chilli peppers (though the latter is hot for a different reason - mustard oil as opposed to capsaicin).

So it depends on what you mean by spicy. A curry might be spicy, as in tasting of spices, but still be mild in terms of heat.