Hughes's Birthday Letters as speaking of the pain that accompanies any relationship.  Many of the poems operate in this context, outside of the specific conditions that entwined Plath and Hughes.  For example, in "Epiphany," Hughes speaks of the hollowness within the decay of marriages, in general:  "Is what tests a marriage and proves it a marriage-/ I would not have failed the test. Would you have failed it?/ But I failed. Our marriage failed."  The idea of failure in marriage is something that can be understood by many and not limited to solely Hughes and Plath.  Another example of a poem that speaks to the pain intrinsic to love's death can be seen in "Life After Death," when Hughes writes of the howling of wolves to communicate something profound:  “The wolves lifted us in their long voices./ They wound us and enmeshed us/ In their wailing for you, their mourning for us."  In these examples of poems from Birthday Letters, Hughes speaks of the pain involved in relationships that fail.  That is a universal experience. Certainly, knowledge of Plath's and Hughes's own failures in love adds another dimension to it.  Yet, it seems to me that there is a particularly universal poetic feel to the poems featured in Birthday Letters, works that seek to express the pain involved in seeing love bloom as a garden only to wind up as a desert.

At the same time, I think that knowledge of the Hughes- Plath relationship adds a dimension to the work featured in Birthday Letters.  It is evident that the death of his wife left a scar on him.  When he writes about how the eyes of their son are “wet jewels,/ The hardest substance of the purest pain/ As I fed him in his high white chair" or how  “his sister grew/ Paler with the wound/ She could not see or touch or feel," it acquires greater meaning knowing the details behind the marriage between both.  Hughes's description of their children is magnified when one recounts that on Plath's final morning, she prepared their milk, and then turned on the gas to the stove in order to kill herself.   Knowing the background of Plath and Hughes, his description of their children acquires greater meaning and significance.  The "paler" nature of their daughter is reflective of Plath, herself, a reminder that something of her lives even if she does not.  I think that this is one instance where the knowledge of the relationship adds to the poem.  

In "The Blue Flannel Suit," Hughes describes some of his first interactions with Sylvia.  Lines like "the lonely/ Girl who was going to die" and how Plath wore "That blue suit, /A mad, execution uniform" are direct references to the cultural image that Plath bears in the poetic consciousness.  They are lines that serve as allusions to Plath's own work in writings such as The Bell Jar.  Knowledge of their relationship enhances such poetic verse.   Hughes's ending of the poem is enhanced through recognition of the relationship he shared with Plath:

As I looked at you, as I am stilled 
Permanently now, permanently 
Bending so briefly at your open coffin.

The closing to Hughes's poem is an allusion to the ideas that Plath illuminates in works like "Lady Lazarus."  It is in this light where knowledge of the relationship between both helps to increase a level of appreciation in the poetic construction.

Another dimension to this would be how Hughes has been viewed in light of Plath's death.  There are varying opinions to Hughes's actions in what he would call "being auditioned/ For the male lead in your drama.”  In some circles, Hughes is excoriated by what is seen as his abuse and betrayal of Sylvia, knowing very well her fragile emotional condition, actions that some believe brought on her suicide.  The publication of Birthday Letters can be seen as a response to these voices, a way for Hughes, himself to offer justification behind his actions in the relationship.  Published months before his own death due to illness, Birthday Letters can be seen as a way to offer dialogue and perspective to a relationship that many have poured over, one where blame has been assigned quite often and frequently to Hughes himself.  In doing so, Hughes might be demonstrating a universal precept that nothing is clear and absolute when it comes to the dissolution of emotional affairs.  In this light, the publication of Birthday Letters gives more potential discussion points, but little in way of answers.  Once again, this is more reflective about the general nature of love and its decay than anything else.

I am not sure if there are specific "advantages" or "drawbacks" in terms of knowing Plath's and Hughes's background.  I tend to think that both are complex thinkers and were immersed in a complex relationship.  The poetic constructions of each speak to that.  Their love and their poetry were parts of their lives.  In knowing about their backgrounds and the relationship that played such a large role in defining they and their work, there might be context given to some of the work.  I don't think that engagement in the poems in Birthday Letters is closed off if one does not know about their relationship and background. Poets like Plath and Hughes construct their work with their lives in their mind, as would any artist.   Artists might strive to keep their life out of their work, but the creative process does not take place in a vacuum outside of life.  In the end, we have more data points, more information.  The search for absolute answers probably eludes us, just as it did Plath and Hughes.