Really interesting question. I'll try give you some of my thoughts.

My knowledge of the crusades is largely confined to the 1st, and a little on the 4th. But generally, you get a sense that it was not technical and tactical innovation that led to success, but sheer volume and zealousness that ultimately scored victory - particularly in the first. However, this is difficult to pin on merely religious belief. In reality, and as many historians will suggest to you (in particular, Christopher Tyerman discusses it at length in his book, 'A God's War'), the average man had far grander visions of success on crusade that went far beyond religious motivations. You just have to look at individuals that led the 1st Crusade, such as Tancred, whose chronicler wrote of him storming a Christian church, and declaring he was satisfied with the amount of 'booty' he could procure from it.

This is straying from the point of the question however. realistically, there was little tactical or technological advance during early crusades. The persistent use of siege tactics, such as at Acre, particularly showcase that old established tactics remained in use. Arguably, however, zealous beliefs did have an impact. The final battle at Acre was supposedly turned around after the holy lance was discovered on site (it's mentioned by a couple of chroniclers, but frankly, a random stick could have been picked up!), which rallied the troops. In essence, I would argue that crusaders did not need to innovate - they had the weight of force behind them to storm to victory.

To answer the second part of the question, innovation I believe is largely linked to the pace of the war in question. Scottish wars of Independence is an interesting one, as it was a fast moving war in which one side (the Scots) were at a particular manpower disadvantage. This is where we see tactical changes, in order to address specific battlefield problems. The rise in use of smaller, more concentrated forces was a direct result of the Scots trying to use the land to their advantage. The best example is perhaps the Battle of Stirling Bridge, where the classic English tactics of charging cavalry followed by overwhelming infantry were thwarted by pike use in a concentrated area (the bridge itself). Tactical development I believe is a product then of specific battlefield problems becoming apparent in a fast-paced war.

Technological advancements I think are more down to the prospect of arms races. The American Civil War is a good example of this. James McCarthy goes into detail in 'Battle Cry of Freedom' over the topic of 'ironclads' - essentially iron battleships that started to replace wooden-hulled ships of the time. Whilst arguably they played little role in the war, their development started as a result of the North becoming aware of French examples, and the South swiftly followed suit in building there own, although on a much smaller scale.

Again, the American Civil War was brutal and fast-paced, and as a result we see tactical evolutions as well. There was a move away from slowly advancing lines of musket fire towards smaller dash groups of men looking to slowly break such lines - a more sort of guerrilla style of fighting. This is particularly evident in the later years of the war, as breach-loading arms and rifles became increasingly prevalent.

Hope that gives some insight!