The dictionaries treat them to be the same, so there's no need to worry about their difference.

Besides, these are old, uptight sounding words falling out of favor nowadays.  They used to be common everyday words, but in modern Korean 여쭤 보다 and 여쭈어 보다 are probably the only common forms still in use.  I think using 여쭈다 or 여쭙다 by itself is rare these days. 

In most cases, people use 묻다, 물어 보다, 여쭤 보다 for asking a question and 얘기하다, 말씀드리다, 인사드리다 for notifying or greeting where you would have used 여쭙다 or 여쭈다 traditionally.

As Tony taught, there is no difference between 여쭈다 and 여쭙다.

It's basically same meaning with 물어 보다.

But 여쭈다, 여쭤보다 is more polite way to use.

You can use this words for elderly people such as a grand father, a grand mother.

But when you use 여쭈다, 여쭤보다, try to say 여쭤보다, 여쭈어 보다 not 여쭈다.

여쭈다 sounds very old fashioned.

Cheers.