They both can mean "to receive". It seems as if 受 has a more negative connotation than 收 but I'm 100% sure about the exact usage differences.
Considering 受 and 收 as the translation of the English verb "to receive" is confusing, as it really depends on how they are used (a lot of expressions use those two words). I think the difference is easier to see with examples.
受 seems to be used for immaterial things (such as feelings, influence, etc.) Its passive connotation really depends on the context or other words in the sentence/expression.
- 他很受欢迎。 - He is very popular.
- 他受到你的影响。- He is influenced by you. (which can be good or bad).
Even when the character 受 is used in words, it is for abstract things: 感受 (a feeling)
收 seems to be used for material things (a letter, a document, a SMS). ex:
- 我收到你的短信。 - I received your SMS.
- 别忘记收垃圾。- Don't forget to collect the trash.
Same things here, when 收 is used in words, it describes material things: 收音机 - Radio (set). Literally the device receiving sounds.
I hope this answer your question. I wish you pleasure and success in your studies!
① 把外面的事物拿到里面；把摊开或分散的事物聚拢：收拾 | 收藏 | 收集 | 收篷 | 衣裳收进来了没有？
② 取自己有权取的东西或原来属于自己的东西：收回 | 收复 | 收税 | 没收。
③ 获得（经济利益）：收入 | 收益 | 收支。
④ 收获；收割：收成 | 秋收 | 麦收 | 今年早稻收得多。
⑤ 接；接受；容纳：收报 | 收留 | 收容 | 收礼物 | 收徒弟。
⑥ 约束；控制（感情或行动）：收心 | 我的心情像断了线的风筝似的，简直收不住了。
⑧ 结束；停止（工作）：收工 | 收操 | 收场
① 接受：受贿 | 受教育 | 受到帮助。
② 遭受：受灾 | 受批评 | 受委屈。
③ 忍受：受不了 | 受得住。
④ 适合：受吃（吃着有味） | 受看（看着舒服） | 受听（听着入耳）。
tl;dr: 收 takes more concrete objects than 受. Moreover, the exact senses of "receive" that the characters can respectively take on are not identical.
My perception is that the difference between 收 and 受 is not directly linked to either:
Whether the connotations are positive
Whether the speaker is passive or active
I will first give examples of when we use each character and indicate why these explanations do not quite work, before giving some suggestions which I believe might better capture the difference.
Examples of using 受
受 does often appear in situations where we might translate it as "bear" or "shoulder" (which in English generally accompany negative statements, not positive ones), and where the subject is relatively passive. For instance:
虽然她小时候受了很多困难，她竟然能够成为一位有影响力的学者。 (Even though she faced many difficulties in her childhood, she was still able to become an important scholar.)
I also should note that in these cases, "receive" is an inappropriate translation. However, 受 can also appear in relatively neutral or positive situations, and the subject can sometimes be "involved" in the action. Sometimes, "receive" might be an acceptable translation. Example:
受了训练之后，我说日语的时候发音发得更准。 (After [receiving] some training, I had more accurate pronunciation when I spoke Japanese.)
我受我朋友影响，每星期天去教堂。 (As a result of my friends' influence, I am going to church every Sunday. A more literal, but perhaps less ideal translation: Exposed to my friends' influence, I am going to church every Sunday.)
Examples of using 收
There are also instances where 收 can accompany unequivocally negative scenarios, or where the involvement of the subject is minimal. Modified example from a news article:
她曾经收过死亡威胁。 (In the past, she had received death threats.)
This is in contrast to scenarios where the subject is fairly involved, or where the connotations are neutral or positive. Example:
我今天收到你的毕业证书。 (I received your diploma [earlier] today.)
Notice also that "receive" is more consistently a good translation.
What then is the difference?
受 tends to be associated with more abstracted objects. It seems to generally mean something along the lines of "to experience", "to be subject to", "to be exposed to". A more accurate gloss is often possible, but may not be generalisable to all instances where the character appears.
收 often indeed means "receive", although "collect" is at times more appropriate. However, 收 is only appropriate when there is an "object" to be obtained, something relatively "concrete" (e.g. a signal for my phone, a letter, money). It is generally inappropriate when by saying that I received something, I mean that I was subjected or exposed to something (e.g. a blow to the head, an education at such and such a university). This more or less corresponds to what I describe as more "abstracted" objects.
Finally, I should note that in some cases, such as in the death threat example, both 收 and 受 could conceivably be used with an object. However, the word chosen in these cases reflects the emphasis of the clause, and the exact usage may be different. I would read "他收到实物威胁" as simply indicating that death threats were received. In contrast, I would read "他受到死亡威胁" as indicating that the death threats influenced my subject somehow, and moreover, I would see the fragment as incomplete (e.g. here, as part of "他受到死亡威胁，离开了欧洲").