as for me, both are the same meaning, but "어쩔수 없죠(뭐)" is more common and natural in spoken. it can be translated as "so be it" or "i can't help it" or "these things happen" depending on context. in my two cents, '~어요' or '~네요' makes sentences tender and more polite, and women or children prefer to use these expressions.

"어쩔 수가 없어요." "어쩔 수가 없네요." They mean almost same. You can use both when you need to say like that "I just can’t help it." When you don't have a way to do so There isn't any way. O.K to use both. But more excatly, they have a different nuance in conversation. "어쩔 수가 없어요" : telling the truth with some of your feeling about no way situation. "어쩔 수가 없네요" : telling the truth with lots of feeling more than "어쩔 수가 없어요" The feeling which i'm saying is kind of "aching sorry, a bit sad" Because you don't have a way. Even though you want to solve, give the way. In this situation you can't. So you feel that feeling. I wish My commet is helpful for your question.

There is no way (to do it, to solve it, etc) I can't find a way (to do it, to solve it, etc) 어쩔 수가 없어요 & 어쩔 수가 없네요 mean almost same to each other, But the nuance, as in Korean & Japanese language , is slightly different ~네요 means slightly more 'sorry' about a fact , but it depends on the context, (i'm talking about only the sentence you wrote) example for ~(어)요 , ~네요 비가 와요 - just a statement, but can be poetic , lol 비가 오네요 - slightly more poetic, d epends on person, IMO 한국이 졌어요 - just a statement 한국이 졌네요 - feeling sorry about the fact, or , ironically , it can sound like sacarstic, depends on the context don't be so confused, basically two endings make same meaning, but its nuance is slightly different depending on the context (that's why it's hard to 'master' Korean & Japanese , but unless you are going to be a native speaker)