Learn English – Difference between “Why did she…” and “Why would she…”

Firstly: Both of them mean essentially the same thing. He had a problem, in the past. He does not have the problem now. The difference is in how the time of the problem relates to other time periods.

"Had" is called the simple past. He had a problem, then the problem went away. This is very neutral on things like how long the problem was there, or when the problem happened, compared to other things the speaker says next.

"Was having" is called the past continuous. Anything the speaker says next is likely to be about things that happened at the same time as the problem.

Secondly: Both of your example sentences should say "the same problem as", not "same problem like". "The same [X] as" is the standard English structure here. (I used it myself in the previous paragraph!)

In many contexts, the meanings are pretty much the same, but you might hear finished more often than completed in casual conversation. For example:

I've finished my shopping.
She finished the song.
He finished the race.

I could use completed in those sentences – the meaning wouldn't change, but the register might sound off.

The word completed can convey some sense of accomplishment. In the context of a race, it might work when the race is a major achievement:

He completed his first marathon last year.

Homework, though, is not really a major achievement, so I think you'd hear finished more often in casual conversation:

“Joey, where are you going? Did you finish your homework?”

That said, you might see completed in more formal contexts, such as a paper on education, or a course syllabus:

Students must complete six homework assignments during the semester.