1. 亲自 is usually an adverb, while 自己 can be an adverb as well as a pronoun mostly.
2. In the example you show here, 亲自 and 自己 are both adverbs, which can be translated as "by oneself/on one's own". They have similar meaning here.
3. Sometimes 亲自 is quite different from 自己 when they are both adverbs. 亲自 just means "by oneself/on one's own"(to emphasize one takes the action and the action may be very important), but 自己 can also mean "alone(to emphasize no others take the action with you)". Here are some examples,
a. 老师问小明:“平时谁送你上学?” 小明回答说:“我自己来回,不需要家长接送。” ——Here 自己 means "alone" and if you replace it with 亲自 the sentence will be a little odd.
b. 这次的生意对公司很重要,总经理要亲自/自己跟对方谈判。——Here 亲自 and 自己 are both OK. If you use 亲自 the sentence means the business is so important that the GM has to negociate with the customers on her own and she may take some asistants with her. If you use 自己 the sentence means the GM thinks herself as the only suitable person to conduct the negociation and she don't need others to help.
4. When you use 自己 as a pronoun, it means "oneself" or "one's own" with "我、你、他、她、他们、它" before it.
a. 我恨我自己。 ——I hate myself.
b. 这是我自己的事,你别管! ——This is my own affair and don't interfere!