Best time to visit

You will likely find little to no difference in costs.

Deends on how you travel.

Transportation-wise the Bahn fares are not seasonal, although if you want to use Saver Fares, you might get a little better price in the spring, when not so many people are using them. Regional passes like the Bayern-Ticket are always available for the same price.

Summer starts about June 21, so late spring is May and most of June; early summer is last week of June and July. Many properties have higher prices in Hauptsaison (high season) than they do in Nebensaison (low season). Whether they change prices, when, and by how much depends on the nature of the town (resorts have high seasons), but I would say a lot of places are already in high season by May. But the later you go, the more likely you are to be in high season. I've also noticed that the small mom 'n pop places are more likely to charge the same price year round, while the fancier, more expensive places (like chains) are more likely to charge more for high season, special events, etc, so sticking with small places is more likely to be budget friendly.

Earlier can be cheaper, but the weather won't be as good.

To get the cheapest hotel or Pension rates, if they are offered by the establishment at all, then you should travel in end July/August. I was told this specifically by a hotel manager in Vienna when I went in to check the place, a three star, and asked if the hotel gave out a brochure. This most definitely applies to Austria and Germany. It all depends on the city, the establishments, the value, etc.