There is speech level difference between 보고 싶다 and 보고 싶어.
보고 싶다 is Haera-che
보고 싶어 is Hae-che
There are seven speech levels in Korean. https://en.wikipedia.org/wiki/Korean_speech_levels
This conversational style is generally called the "plain" style. In writing and quoting, the plain style is the equivalent of the third person. Any other written style would feel like a first person account (that is, anything else would seem to be told in the main character's own voice). It is used:
To close friends or relatives of similar age, and by adults to children.
In impersonal writing (books, newspapers, and magazines) and indirect quotations ("She said that...").
In grammar books, to give examples.
In some exclamations.
This conversational style is called the "intimate" in English. It is a defective paradigm, meaning it lacks most of the expected conjugational forms. It is used
Between close friends and relatives.
When talking to children.
The hae-che and haera-che styles are frequently mixed together in the same conversation, so much so that it can be hard to tell what verb endings belong to which style. Endings that may be used in either style are:
~다/하다 of Hae-che is actually a kind of conjugation.
In case of adjective, it is same to infinitive.
In case of verb, it is not same to infinitive.
조용하다 -> 조용하다.
좋다 -> 좋다
공부하다 -> 공부한다.
먹다 -> 먹는다.