In the real world the Olympic Games were originally an exclusively Greek affair but with the Roman conquest more and more Roman athletes began participating. Asterix suddenly points out that the Roman conquest means that the Gauls now are Romans. I'm not entirely convinced that this interpretation of both Roman citizenship laws of the time and the Olympic participation rules would have got passed real world lawyers (and the idea that the inhabitants of a place that has opted out of a continental spanning political entity can still claim the rights and privileges of its citizens is an interesting one which naturally brings to mind the current Brexit negotiations, but this isn't the place to go into that). However, in the story this interpretation is not challenged and thus all the men of the village set off to Greece to support the participation of Asterix and Obelix.
This is a very different story from the norm, with no grand battles between the Gauls and the Romans. Even the pirates escape conflict on the first meeting, though they still scuttle their ship, and the second encounter is covered by a single panel showing the aftermath. There are a few individual moments when rival athletes are knocked away but otherwise the fighting is limited to the wrestling contests. Instead the emphasis is on both competitive sport and tourism, with the links between the two explicitly acknowledge when the Olympic officials opt to create an extra event exclusively for Roman athletes in order to maintain international interest and tourist income. As in the real world, sporting victory brings prestige not just to the athletes and their trainers but also to their homes, making the contest a peaceful way for intra-Empire rivalries to assert themselves.
The tourism industry is mocked heavily with both dubious bookings and nepotistic arrangements highlighted. The Gauls travel to Greece on a galley that has "One class only, deck games, open air sports and marvellous atmosphere". It turns out they have to row themselves to the atmosphere of drums beating. When they arrive at Athens they are soon taken up by the tour guide Diabetes, who ensures they can find hotels, restaurants, taxi buses and bureaux de change - all run by cousins of his. In general, the Gauls wind up playing the role of stereotypical disrespectful tourists, cramming into hotel rooms, bringing their own food, making a spectacle of themselves at nights and being rowdy in the stadium, in stark contrast to the stoic approach of the Greek spectators.
Much of the contest with the Romans winds up a struggle of will and determination, with the local Roman athlete Gluteus Maximus initially extremely confident of victory until he encounters Asterix and Obelix in the forest and they outperform him in evert way. This results in a crisis of confidence, broken only when he believes he won't have to compete with the Gauls, until eventually his centurion Gaius Veriambitius gets a ruling on the magic potion. This story is especially contemporary in its handling of the issue of drugs and other enhancing substances in sports. In the real world the 1968 Olympics were the first games to take place with a ban on such substances, with the first athlete to fall foul being Hans-Gunnar Liljenwall, who drank two beers to steady his nerves. Although the real ancient games had involved many substances, here the rule is back projected and thus the Gauls' hope that the magic potion will bring them easy success is soon thwarted whilst the Romans are warned off alcohol. Without such enhancements, Asterix and the Romans lose to the Greek athletes every time, though it's notable that Asterix is still shown as faster than the Romans. But even this is not enough to give confidence of victory in the "Romans" only event and so he and Getafix concoct a cunning plan to take advantage of both the rules and the magic potion.
This solution is entirely in keeping with the depiction of Asterix as a man who succeeds through cunning and not merely because he can quickly magnify his strength to overcome his foes. But normally he is in a situation where there are no fair rules and he has to survive by his wits end. Here he is taking part in a long-established tournament with clear rules of contest and setting up his rivals to get them disqualified is undoubtedly a form of cheating, even though it's the Romans who commit the actual offence themselves. This doesn't feel quite within the rules of sport and perhaps explain why Asterix opts to give away his palm of victory, though he explains to Getafix "I gave it to someone whose need was greater than mine" and asks the druid to not repeat this. Although it's not dwelt on, the earlier races had shown Asterix beating at least the Romans and so it's highly likely he could have won the race fair and square anyway. As a result, the ending of this tale feels somewhat off, undermining the hero's physical prowess and making for a highly unsatisfactory ending.
It's a pity that the story ends in such a way as it's previously been a marvellous satire on tourism and sport, making many good points through humour. Unfortunately, by focusing on the hero's cunning instead of his physical abilities even without enhancement it results in this dissatisfactory conclusion.