Please reply by tomorrow.
I look forward to your continued support.
お願いします is the standard polite request formula, while お願いいたします adds the humble いたす for a more deferential, formal tone used in business or service settings.
ください directly attaches to a verb to mean 'please do ~', while お願いいたします is a standalone humble formula often used with nouns, as in ご確認をお願いいたします.
Standard mixed spelling for this polite humble request formula; writing いたします in kana is common in modern formal prose.
Common kanji variant using 致します; it can look slightly more kanji-heavy or formal than お願いいたします.
Formed from お願い, the polite noun form of 願う 'to request, wish,' plus いたします, the humble polite form of する 'to do.' The phrase functions as a deferential request formula.