What causes an inability to recall a melody, beat, or music? Background:
I have never been able to recall melodies to songs. I can play a song over and over again but the moment it stops, I cannot remember what it sounds like. I am referring to the music/beat/melody and not the lyrics. Even when I memorize the lyrics and try to sing a song out loud or even in my head, I simply cannot recall the music.

I like Adele but as I type this, I have no idea what her songs sound like. If I play one of her songs, I will recognize it. So it is not as if I have lost all memory of the song.

Over the years I have come up with certain tricks to remember some very rudimentary melodies. I know what Twinkle Twinkle Little Star, the happy birthday song, Jingle Bells sound like but it took work. For Jingle Bells, I would knock on a table as it played. And I was able to get the rhythm. The lyrics to Twinkle Twinkle and the birthday song flow like a poem and so the melody strives from that  (at least that's how it seems to me).

Even though I have no idea what my favorite songs sound like, I do know what emotions they evoke. When songs move me, it is as if something flows in me. I have heard stories from my mom about how I used to close my eyes and sway when listening to music. I learned not to do that in public after the teasing at school :)

I have tried to research this on my own and have come up with nothing. Any insight is greatly appreciated. 

Question

What causes an inability (or difficulty) to recall a melody, music, or beat?
The following answer is based on my own experience learning music combined with general principles of cognitive psychology related to skill acquisition.

I think that learning music would help a person recall a melody, a beat, and music in general. Formal training would be particularly helpful, but informal training would also often have a similar result.

In particular, learning music teaches you how music is structured. This training teaches you a whole range of skills. You learn to isolate instruments. You learn about the structure of songs (e.g., bars, beats, sections, repetition). You learn about melody (e.g., scales, intervals, chords, progressions, etc.). You learn about rhythm. 
There is a theoretical aspect to this knowledge whereby you conceptually understand the ideas. However, there is also a large procedural component whereby you train your ear to identify different sequences, and you also learn to produce such sequences with a suitable instrument.

Learning to sing can be particularly helpful in training your ability to reproduce melody without the song present. Singing trains you to be able to reproduce intervals, scales, and so on without an external source.

All this declarative and procedural knowledge of music simplifies and enables the mental recall of music. For example, instead of seeing a pop-song as a 3 minute stream of sound, You start to see the simplicity of for example, 3 chords, and 2 major melodic elements. You see the similarities with other songs in terms of the chord progressions, scales, and rhythm. It's similar to the way that language becomes simpler once you appreciate the basic principles of vocabulary and grammar.

Beyond musical skill, actual repetition and spending time seriously thinking about the song should also help in the recall a that specific song.