Usage differences of “obwohl” and “trotzdem”

Per Duden trotzdem is an adverb. In the German language, some adverbs are also used as connectors, for example, trotzdem, jedoch, deshalb, deswegen, etc. However, when these adverbs are used as connectors, the verb position must still be placed at position 2 instead of at the end (this differs from conjunctions such as obwohl, weil, damit, ...).

This use of adverbs as connectors does not exist in English language. In English, only conjunctions can connect sentences with a comma and never only an adverb.

The correct sentences are:

Clarissa bleibt nicht im Bett, trotzdem ist sie krank.

--> verb cannot be at the end.

Clarissa bleibt nicht im Bett, obwohl sie krank ist.

--> verb must be at the end.

Trotzdem and obwohl do not have the same meanings: trotzdem: nevertheless. obwohl: although.

In case of trotzdem, the conjugated verb stays at 2nd position as it's mostly used adverbially, but in case of obwohl, the conjugated verb goes in the end.

For example:

Obwohl ich krank bin, bin ich in die Schule gekommen. / Ich bin in die Schule gekommen, obwohl ich krank bin.

Ich bin in die Schule gekommen, trotzdem bin ich krank.

(Note that in case of trotzdem, the position of conjugated verb is at 2nd position. Additionally, it cannot be written the other way around.)

This is a somewhat uneasy subject to digest. I try my best to explain based on the Duden.

Taking your example the first is fine and obwohl can be replaced by trotzdem; though obwohl is more often used and considered high-level language. In this case obwohl and trotzdem are conjunctions. The usage of trotzdem in this case is considered colloquial language (umgangssprachlich).

Clarissa bleibt nicht im Bett, obwohl sie krank ist.
Clarissa bleibt nicht im Bett, trotzdem sie krank ist.

The second sentence is correct. Though many German speakers would say that it sounds weird to them.
I suggest you use obwohl when not sure and because it is not colloquial.

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There is another meaning of obwohl, which is colloquial, and cannot be replaced by trotzdem.

Ich rufe Dich heute Abend an. Obwohl, wir sehen uns ja morgen ohnehin.
I call you this evening. Though, we'll see each other tomorrow anyway.

Here it is used to say that Maybe I call, maybe I don't; knowing that we see each other tomorrow anyway.

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The word trotzdem can also be used as an adverb. That can be confusing for speakers. This is how you used the word trotzdem in the second example and in this case it is wrong by meaning; you are not saying what you want to say.

Clarissa bleibt nicht im Bett, trotzdem ist sie krank.
Clarissa doesn't want to stay in bed, nevertheless she is sick.

When trotzdem is used as an adverb it does not have the same meaning as obwohl. In this case trotzdem could be translated to even so, anyhow, nevertheless, ...

First, your grammar, in particular your word order is correct. But I think that the meaning of the second sentence is not what you thought. The first is ok.

A, obwohl B

means that both A and B are true, and it points out that in particular B did not prevent A from being true, even though it should in general, or one might think that it does, or it usually does, or something.

Your first sentence thus means: Clarissa is ill. She did not stay in bed. It also conveys that Clarissa should have stayed in bed, because she is ill, or maybe, that it is commendable that she did not let her illness prevent her from getting up.

The same is expressed by:

B. Trotzdem A.

So the above meaning could also have been expressed by:

Clarissa ist krank. Trotzdem ist sie nicht im Bett geblieben.

(Clarissa trotzte also ihrer Krankheit, indem sie nicht im Bett blieb.)

Your second sentence, on the other hand, says that Clarissa is ill and did not stay in bed, and that that is surprising, since usually getting up prevents you from getting ill. Or, more reasonably as chirlu pointed out, it emphasises that she is really ill, even though her not staying in bed might indicate that she is not.