I am at beginner-intermediate level of Japanese, so please do not consider this as an Authoritative Answer having precedentiary value, it is completely based on inference derived by me however, in support I have presented my arguments in support of my answer.
So, please correct me, if my reasoning is wrong. Thank you !!
It is clear that でしょう when used with rising ending intonation, it would mean, that the person is seeking for hearer's agreement, on something. E.g. 君も行くでしょう? (You will go, won't you?)
Whereas, ね is also used to get the hearer's agreement, with the intonation is falling one.
ね of Agreement and ね of Confirmation
Now before we get into the difference we should first understand what, actually this agreement ね is!, as ね can also mean seeking for confirmation (Intonation is rising), which would mean "I assume XYZ can you confirm it?", whereas, agreement ね is a speaker's strong impression of something, and he wants the hearer to agree on it, it is not seeking for confirmation, but agreement (hereinafter for the sake of brevity referred to as, agreement ね).
E.g. あなたは学校生ですね (with rising ね can mean, that the speaker assumes that the hearer is a student, like he saw a school bag/clothes and assumes it, and wants to confirm it.
Whereas, one school student asks another student to rent a motorcycle, and the other can reply by saying まだ学校生ね. (This form of ね would mean, I am still student, right?). Here it can be seen that, this case is not that the person wants to assume he just wants an agreement.
ね of Agreement as per me
Now, coming to the main point, this 2nd ね, the ね of agreement, as per my book (A Dict. of Basic Japanese Grammar), is used when there is some shared knowledge.
My Inference, only, from hereon: Please don't take it as an Authoritative text
That is to say, both should have the same knowledge/experience on the act/scenario/situation (topic of the conversation). That is, both should have experienced the same and have a common knowledge, such that both the person are able/in a position to relate it.
E.g. A and B are walking together, they see a cute dog. Now, in this scenario, both/either can use the seeking-agreement ね and say かわいい犬ね? As, the other person has also seen that, i.e. there exists a shared knowledge/experience which was gathered by watching that dog 犬 and the other can now relate it.
Whereas, If B over here did not see the dog 犬, A cannot say, かわいい犬ね? (Using agreement ね) as, B cannot relate which dog/or what is A talking about.
Further, as per me it is not necessary that the other person has in reality, i.e. literally, experienced the same situation/action/scenario, using ね in such situations, can mean that you can relate to it even without actually experiencing/having knowledge about that.
E.g. A tells B, I was not able to go to Japan to witness the New Year's celebration. Wherein, B replies, 残念ね (with agreement ね), and A replies 残念です. Over here, if we look at the scenario, it was not that B was not able to go to Japan for the New Year's Celebration, or even in first place planned to. So here, we can see that B tried to relate the situation, in which A was in, in order to empathize with A. (In short saying I can understand, I feel bad for you).
However, as I can't come up with more examples, as I am still at budding stage, so I am unclear to what extent this "Trying to/Able to relate with no experience/ knowledge scenario/action, by using agreement ね, applies".
The points below are purely out of inferences derived from looking at various sentences using the agreement ね:
Moreover, as per my inference, this scenario/action/situation in relation to which agreement ね is used, can be in past, whereas, the statement for present, in relation to that past scenario/action/situation. E.g. A and B, are planning to a dinner, where B says we will eat a lot tonight, and when they go out in the night, B before starting the dinner can say たくさん食べようね (using agreement ね), over here we can infer that, such usage of ね, is can imply there that there exists a relation to the shared knowledge about the past scenario/action/situation (which is, the conversation, i.e. both of them have this shared knowledge/experience).
Thus, as per my inference, agreement ね can only be used, when the speaker and the hearer, both of them find a relation to the shared knowledge/experience of a action/situation/scenario.
So, I suppose, though I am not confident at all, that only in case that they can relate it with a shared knowledge/experience of a action/scenario/situation in regard to which statement has been made, then agreement ね will be used.
Now, coming to でしょう, the rising intonation, at the end, tells us that the speaker is asking for agreement (hereinafter, agreement でしょう) and not that he wanted to mean "probably".
As per, Misaさん
Agreement でしょう, can be in those cases instead of the Agreement ね, when the speaker is having prior experience of a thing, and when the speakers makes the hearer experience the same, agreement でしょう can be used. E.g. A and B go to the restaurant, where A knows and has told B that tasty food is available, so when both eat, A can say to B, おいしいでしょう? (with agreement でしょう). Click on the link, there are 3-4 other examples referring to same scenario.
My own Inference derived from the same (Not sure):
Now, if we look at the above example, we can clearly see that, since the speaker focused not on the point of shared knowledge/experience which they acquired when both ate the tasty food, and on the fact that prior to that, i.e. him knowing that the food was tasty here. Thus, it can be inferred that in those cases where there isn't any shared knowledge/experience of a action/scenario/situation or it is not the intention to bring it up, でしょう would apply.
(Not Sure- Self-made example)
E.g. A Husband, suddenly plans to go to a Wedding Anniversary, and informs him about the wedding and says, 君も行くでしょう, over here as shared knowledge/information with the wife that she would have to go, as per my inference, でしょう would be used, as he really wants his wife to accompany him.
Coming to the Question
As per the above-based inference and reasoning of mine, which I am still not sure about, 疲れたね, could have been used in those situations wherein, there exists a shared knowledge/experience about a action/situation/scenario to which one can relate. E.g. A and B work together in the same shift, and there shift ended after 7 hours, so over here, since A can relate how much tiring it was as he did the same, he can say to B (who also worked for the same time), 疲れたね (with rising intonation).
Also, note that over here if A was not working with B, but was elsewhere and then he would have said this to B, this ね would have been the ね of confirmation i.e. I am assumed you are tired after working for 7 hours, can you confirm it?.
Whereas, 疲れたでしょう would have been used in a situation where the でしょう is of agreement, where the person cannot relate to the action/situation/scenario pertaining to the work/tiredness that the hearer might have that he has. E.g. A a software engineer, who is a friend of B (the person who did physical work for 7 hours), insists that B comes to drinking sake after work with A, and says, 君は疲れたでしょう, to convince for a drink saying him release his tiredness (which A has no account of/can relate), by drinking alcohol.
I hope it might help ! I know these are more of inferences, however, I tried to give a wide perspective, and the possibilities that it might have.
Click hyperlink Misaさん, she has tried to explain the difference. However, I felt that it needed more explanation.