'though', 'although' and 'even though'

Though, although and even though are used to show a contrast between two clauses:
Our new neighbours are quite nice (this is good) though their two dogs bark all day long. (this isn’t good)

We can use though or although with no difference in meaning. But, some differences are:

Though is more common than although in conversation or writing.
Though (but not although) can come at the end of a sentence:
My new bike is really fast. I don’t like the colour, though.
Though (but not although) can be used as an adverb:
I’m not good at maths but I can help you with your geography, though, if you want.

The meaning of though is similar to however, but though is much more common than however in conversation.

Even though can be used to make the contrast between two clauses stronger:
Dad got back from work really late, even though he had promised to take mum to the cinema.

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They are similar in meaning, yes. They all serve to record something that is surprising or unexpected. But the difference in usage is that although, though and even though are all conjunctions, whilst in spite of and despite are both prepositions.

So usage requires:
in spite of + noun
although + clause
despite + noun
though + clause
even though + clause

Although and though can be used in the same way. Though is perhaps more common in informal speech and writing, whereas although can be used in a wide variety of styles.

Compare:
'Our new neighbours are quite nice, though their dog is a bit of a nuisance.'
'She insisted on keeping her coat on, although it was extremely warm in the house as the central heating was on.'
'Although she was commended for completing the Millennium Dome project on time and within budget, management felt that it was now time for a new person with different talents to take over.'

Though is often used with even in order to give emphasis:
'I managed to get good results in my exams, even though I went out four times a week when I was supposed to be revising.'

Whilst despite might be thought more formal than in spite of - it is, after all, one word rather than three - there is really very little difference in usage between the two:
'Despite the appalling weather, they succeeded in walking to the top of Ben Nevis.'
'They decided to get married in spite of the huge differences in their ages.'

So, to summarise: despite and although: similar meanings, but different syntax required. Compare:
'Although it was raining heavily, we finished the game of football.'
'We finished the game of football in spite of the heavy rain.'
'Despite his strong Welsh accent, we understood most of what he was saying.'
'Even though he had a strong Welsh accent, we understood most of what he was saying.'
One further word. Although, despite and in spite of are normally used as prepositions, they can also be used in adverbial constructions with -ing, thus:
'I managed to pass my exams, despite going out four times a week during the revision period.'
'In spite of feeling terribly sick, I went to work every day that week.'
'Despite being severely handicapped, he managed to complete the race.'