Is there a difference in connotation between questions ending in 吗 vs. X 不 X Is there any difference in connotation/region/modernity/etc. between questions ending in 吗 and affirmative/negative questions?
For example, does:

你很累吗？

connote something different than:

你累不累？

And does the following have a third connotation?: 你很累没有？
There's also 嘛 and 吗. Practically, there is little difference between the two, aside from the pronunciation. 

吗 is most used in questions where speaker is usually neutral about the question. 

嘛, although not applicable in your given example, it can be used to question or affirm. I.e., used with to express emotional context (positive or negative given the context) (e.g., 你这是干吗？ = questioning, vs.  你这是干嘛？ = complaining) or to express certainty with something (e.g. 很贵嘛! or 不错嘛！)

In terms of connotation, "你累不累?" is the same as "你很累吗?" They both mean to ask if the person is tired or not, in a neutral tone. 

你很累没有 seems a bit awkward grammatically. The way it's structured implies a statement rather than a question. You are probably looking to use 吧 in this case. E.g., 你很累吧 (You're pretty tired, huh/right/aren't you?), would be more appropriate here. 吧 is used at the end of a sentence when you are uncertain of something you can say. 

There is also 呢, which is often used at the end of an interrogative sentence. e.g., 为什么...呢? 怎么...呢? Take note that 呢 is NOT interchangeable with 吗.