Apesar de que is a synomym for embora, although. Both introduce a subordinate clause that has actually happened and potentially could had prevented another event from happening, but didn't do it: Although he had his arm broken, he drove a car. Obviously there are a lot of variations of this logic. It is possible as well that the fact that an event didn't happen could had been a hindrance: Although she had not invited him, he came to her party. It is possible as well that both sentences are negated: Although he had not broken his arm, he didn't drive a car. The logic however is always the same.
The event described by the subordinate clause happened.
Although he works out a lot, he doesn't become more slim.
The event described by the subordinate clause didn't happen.
Although he doesn't work out a lot, he is slim.
In opposite to although the conjunctions ainda que / mesmo que describes hypothetical situations. In other words it is irrelevant whether the event described in the subordinate clause happened, happens or will happen. That doesn't have any impact on the event described in the main clause.
hypothetical event in the present
Even if he works out a lot, he wouldn't become more slim.
hypothetical event in the past
Even if he had worked out a lot, he wouldn't have become more slim.
Compound conjunctions in roman languages very often suggest a meaning that has nothing to do with the real meaning. In apesar de que we have the word pesar, which means grief, the whole thing together would therefore be something like "in grief that", what has nothing to do with the actual meaning. In a sentence like "Although he is bone idle, he is stinking rich" there is no reason to feel sorrow for him.
Portuguese have taken the heroic decision to use the conjuntivo even with conjunction like embora and apesar de que that describe objective facts and not hypothetical situations. They decided that any concessive conjunction, irrespectivee whether the situation is hypothetical, in other word if ainda que / mesmo que is used, or real, in other words if embora or apesar de que is used. If 420 millions Spanish speakers don't succeed in convincing them that with apesar de que the indicative would be more logical, we are not going to convince them either. (However we can find example with the indicative after apesar de que / embora.)
Furthermore attention is to be paid to the fact that apesar de que is written in one word, not a pesar de que as in Spanish. Furthermore apesar de que is seldom used. (Actually so seldom that a lot of native speakers find that it sounds "strange".)
apesar de que obwohl
É um presente para ti, apesar de que não o mereces.
This is a gift for you, although you don't deserve it.
Gosto de trabalhar, apesar de que isto seja cansativo.
I like to work, although it is exhausting.
More common is the preposition apesar de, without que, used with infinitivo / infinitivo pessoal. (Don't forget: Together with a preposition we can only use nouns or something, like the infinitivo (pessoal), that can have the syntactical function of a noun. Despite in English is preposition and conjunction. But despite as preposition is apesar de and despite as a conjunction is apesar de que. We use an English gerund together with the preposition. The English gerund has absolutely nothing to do with the gerundio in roman languages. The English gerund is a verbal noun, the gerundio in roman languages is never a verbal noun. The English gerund is to be translated with an infinitivo (pessoal). The englisch present participle doesn't differ in its form the gerund, but the syntactical function is completely different. Object gerund: For some people eating is the most important thing in life. <=> Adverb present participle: Eating he told him the whole story.
apesar de despite
Apesar de ser difícil, vou tentar.
Despite being it difficult, I will try it.
Apesar de ser muito difícil, comecei a trabalhar de noite.
Despite being it very difficult, I have started to work at night.
Apesar de saber que isso é impossível.
Despite well knowing that it is impossible.
Apesar dele ser* um grande cantor, eu fiquei decepcionadíssimo.
Although he is a great singer, I was disappointed.
Apesar dele ter* estudado bastante, ele não passou de ano.
Despite being him a great singer, I was deceived.
* For didactical reasons it is useful to understand that it is an infinitivo pessoal. Concerning the form there is no difference to the simple infinitive, since in the 1st and 3rd person singular the infinitivo pessoal has no ending. But we can deduce from the context that the person who is a great singer is not the person who is deceived.