What is difference between yaptıydım and yapmıştım?

-dIydI 

This form is very rarely used in daily language [verbal language] which is very close to -mIştI form.

Verb stem + dI + y + dı + Personal suffix

Yap         + tı   + y + dı    +   m

Bu sabah kahvaltı yaptıydım.

I had had my breakfast this morning.

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Geçen yaz İngilizce kursuna gittiydi.

He/She had gone to English course last summer.

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Dün ona telefon ettiydik.

We had phoned him yesterday.

Çok teşekkür ederim Tunci

simple past tense (-di) + past of additional verb (idi)

-miş + past of additional verb (idi)

-diydi is not used very much in official texts, almost never. But both are often used in spoken.