The word breathtaking is based off of the idea that the sight or experience of something was so amazing or powerful that you were so in awe you forgot to breathe - it literally took your breath away or made you gasp in amazement. It's similar in that way to the word 'speechless'. Of course, the word can be used if you weren't actually breathless. It doesn't have to be used literally, but that is the idea the word is built off of. Thus, it makes sense to describe incredible views of nature and is commonly used that way, but it can be used in other situations that leave one with a sense of wonder or amazement. Brett gave a very good example illustrating the actual neutrality of the word. While 'breathtaking' is typically used to describe good experiences is really more about describing the intensity of it.
Spectacular adds 'dazzle' to the description of something. I don't mean 'dazzle' or 'sparkle' literally, but spectacular kind of sounds like those words and subtly brings those ideas and feelings to mind when building a mental picture. It's a lot like 'wonderful' but with a touch of 'flair'. I would consider it's use to be less intense than 'breathtaking'
Wonderful as you said is a synonym to all these and is almost interchangeable. However, since is is used far more generally and more often it is the less intense of all of the three words.
The view from the mountain was wonderful. (the most general. It sounds nice, but doesn't bring much to mind)
The view from the mountain was spectacular. (A really beautiful view - perhaps the light was dancing off trees- this sentence just feels like it has a little more life to it)
The view from the mountain was breathtaking. (Doesn't bring as much to mind like 'spectacular' did, but it must have been an incredible view for the person. They might have stood there admiring it for longer and perhaps even felt closer to nature and beauty itself)
Each of these sentences conveys a slightly more intense experience than the one before.
They are all by definition quite similar, but I hope this illuminates their use the feelings behind them for you!