Mogę = I may, I am allowed, I am able to (like, able to physically/mentally cope with something):
"Czy mogę iść do łazienki?" - May I go to the bathroom?
"Mogę jeździć na rowerze całymi dniami" - I can ride a bike for (whole) days.
Umiem = I can (I am able to, meaning I know how to do it)
"Franklin umiał już liczyć i sznurować buciki" - Franklin could count by twos and tie his shoes. (from the cartoon, the translation is real and not totally literal)
"Umiem jeździć na rowerze" - I can ride a bike. (I learned and learned and learned and now I know how to)
What about "I am not able to buy everything" as a translation? Wouldn't that express the same idea?
It would, it works now. Just please note that the Polish sentence is general - as in my comment above.
Why not "Nie mogę kupić wszystkiego"?
In this context, is the person saying that he's not a millionaire and can't buy whatever he wants and when he wants (because kupować is imperfective)?
It could be (and it's an ex aequo best answer in the ENG->PL exercise), it was just the choice to use this one in the PL->ENG exercise.
"Nie mogę kupować wszystkiego" generally, like I cannot shop for such a huge amount of stuff every week, stop asking me to!
"Nie mogę kupić wszystkiego" right now, or at least on some one specific occasion.
how come "purchase" is wrong, I thought buy and purchase are similar meaning
Why is wszystkiego in the genitive? I would have thought it would be accusative since kupowac kogo/co?
Yes, but it's negated here. Or rather, "mogę" is negated, which affects the whole anyway.
That's not what the sentence means. The Polish sentence is like "You cannot expect me to buy all this stuff!"
Actually, after your comment we realized that neither "not able" nor "unable" suit this sentence. It uses imperfective 'kupować', so it's not about for example not having enough money on your Saturday trip to the market, it's more like "I already bought you a fur, a new handbag and a necklace, that's enough! I'm not a human wallet!" - and therefore those two seem too weird.