So this is a good question, and one I'd want to point out varies widely festival to festival and location to location. Think how different Thanksgiving in a suburb is from Mardi Gras in New Orleans to Christmas in a small town is today in modern America.

Let's focus on one festival for Dionysus in Athens - The Anthesteria. It was a three day festival with a focus on wine and the dead. The first day was called the Pithoigia (Cask-Opening), the second day was called Choes (Jugs), and the third was Chytroi (Pots).

Pithoigia was a day when the first bottles of the new vintage of wine were opened, and libations were made to the gods. Pithoigia was a happy day, where the enslaved people and servants were allowed to drink (more over, those they served were forbidden from telling them to moderate their drinking). The drinking would continue into the next day, Choes.

Choes was a day of drinking, where three year old boys had a coming of age ritual where they were given a smaller choes (jug of wine, probably their first taste of the stuff), presents, and a garland of flowers. Adult men would share in a drinking competition of downing 3 Liters (About 4 regular sized bottles) - whoever one was given more wine. This was done in silence, for the same reason every drank from their own jug - it was in memory of when Orestes came to the city seeking food and drink - it was given, but it had to be consumed outside and consumed silently, as he had murdered his mother, even if she may have deserved it. This silence was not the only thing solemn about the festival - the day was considered unlucky, and most temples other than the Temple to Dionysus Limnaios (Dionsysus in the Marshes) were closed. This was the day the dead were believed to rise, so many Athenians would chew buckthorn and annoint their front doors with pitch.

But again, the festival was not wholly swayed to solemnity or joyousness - the silent drinking would culminate in private celebrations at home, where more feasting and drinking was done, and when these were done everyone would make their way to the shrine of Dionysus Limnaios, Dionysus in the Marshes, to pour out any wine that remained, and would leave the vessels at the shrined, garlanded in the flower and leaf crowns that had been worn during the day. It is also of note that various women (The Wife of the Archon Basileus, and 14 appointed maidens) met together for a marriage of the Basillina (Wife) to Dionysus, though what shape this took we do not know. Speculations include that the Basileus or a priest of Dionysus played the role of Dionysus, or that a dildo was used, but we really don't know.

Meanwhile, Chytroi appears to have been named for the pots of grain which were offered to the dead. Finally, the festival was closed with the statement: ''Out of the doors, Keres; it is no longer Anthesteria.", which would have ended the festival and dismissed the ghosts to return to the underworld (and would later be used sarcastically to let any mooches know they'd used up all their favors).

So, here, within one festival, we find some elements of joy and partying, but also solemnity, more for the dead or from tradition than for the gods. There are other festivals, such as the Agrionia, which is also a Dionysian festival with connections to the dead (and, Plutarch alleges, perhaps not reliably, origins in human sacrifice), that have a solemnity attached, however the City Dionysia in Athens, though it would have tragedies performed, would be a much happier festival with wine though-out the week and satyr-plays.

Now I'm going to quote from Jane Ellen Harrison here, because she seems to strike a salient point on the difference between solemnity and joy at festivals:

We shall find a series of festivals which are nominally connected with, or as the handbooks say, 'celebrated in honour of various Olympians; the Diasia in honour of Zeus, the Thargelia of Apollo and Artemis, the Anthesteria of Dionysos. The service of these Olympians we should expect to be of the nature of joyous attendance.' To our surprise, when the actual rites are examined, we shall find that they have little or nothing to do with the particular Olympian to whom they are supposed to be addressed ; that they are not in the main rites of burnt-sacrifice, of joy and feasting and agonistic contests, but rites of a gloomy underworld character, connected mainly with purification and the worship of ghosts. The conclusion is almost forced upon us that we have here a theological stratification, that the rites of the Olympians have been superimposed on another order of worship.

- Jane Ellen Harrison, Prolegomena to the Study of Greek Religion

Now, Jane Ellen Harrison is touching on here that she believes that the gloomy, underworld aspect is separate from the joyous, heavenly aspect of the festivals. I wouldn't necessarily agree, but that's starting down a different road. What I think is important to note is that the festivals connected to the dead might be solemn because they are connected to the dead, while festivals unconcerned with the dead and wholly focused on the gods would be more joyous affairs

Sources:

• Walter Otto, Dionysus: Myth & Cult
• Jennifer Larson, Ancient Greek Cults
• Jane Ellen Harrison, Prolegomena to the Study of Greek Religion