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So, 보고 싶다 and 그리워하다 can be translated into 'I miss you'. But since the main meanings are different, those two phrases have different meanings.

1. 그리워하다 = It is to directly saying you miss somebody / something

2. 보고 싶다 = It is just a 'metaphoric / poetic' phrase to say you miss sombeody / something.

However, 보고 싶다 itself doesn't have a meaning of 'to miss'. So, it can't be always translaed into 'I miss you'. It just has a meaning of 'I want to see you (because I miss you.)'. And 'because I miss you' part is usually ignored, since it is clear that why you want to see them. : )

To talk about the difference between 보고 싶다 (by the way, it is 보고(space)싶다 and not 보고싶다. :3) and 그리워하다, the first thing to look at is the meaning of two prhases. : )

1. ~을/를 보고 싶다 = 보다(to see) + -고 싶다(helping verb : want)

 -> Meaning : I want to see ~

2. 그리워하다 = verb itself

 -> Meaning : To miss

That is the main difference between the two phrases. So if you miss somebody, you usually want to 'see' them too. So that is why those two phrases are often used in a similar situations : want to say you miss somebody.

However, unlike 그리워하다 which clearly has a meaning 'to miss', 보고 싶다 doesn't always translate into 'I miss you' all the time. :3 It has its own meanings as 'I want to see~'. So it can also be used in a sentence like this.

영화가 보고 싶다.

(= I want to see a movie. / *영화 = movie) 

꽃이 보고 싶다.

(= I want to see the nature / *꽃 = flower)

As you can see from the examples above, 보고 싶다 can be used in a various situations. But it is usually used in a situation of 'expressioing to miss something / someone' because if you miss sth or sme, you usually want to 'see' them too. Just like the lovesick always has a side symptoms of 'seeing an illusion of someone who you are in love with'. XP

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That answers everything that I want to know!

Plus, I got some correction on the spelling as well. ㅋㅋㅋㅋㅋ

Thank you so much Havina! :D