Japanese usually don't know what the difference between these two words are, and who can blame them? They show up sometimes in literary and seldom in conversation, and they are arguably etymologically the same word.
The root obo- should ring a bell to you. It’s in fact the same thing as omo-. So, 思う, 覚える, and the like all come from the same source with other things added. Japanese had a phoneme sometimes denoted a /mb/ that results in the interchangeability of m and b in many words today. In some cases, distinct words have developed from the same etymological source. This makes 思しき confusing.
Many sources claim that it comes from the verb 思す. This verb comes from 思ほす which comes from 思はす （思わす）. Then, the form 思しい comes from attaching –shi (modern –shii) to the original form of this. This is similar to other words like 願わしい (desirable) from 願う. Thus, 思しい is clearly etymologically the same as 思わしい. One form shows sign of extra sound change and the other doesn’t. Not, surprisingly, natives often don’t distinguish them semantically.
First, we will look at several dictionary endings for the two to see if we can deduce any important information. The first dictionary to look at is the 広辞苑 Edition 5.
Google’s dictionary says the following.
Qualification of Usage
Some speakers try saying that 思しい is only for showing appearance and that 思わしい is for showing what is hoped for. Yet, these entries show that both meanings exist for both. So, we will learn about how they are practically used and grammatical differences that they have to differentiate them.
思しい is going to almost always be used in an attribute phrase and is most often in the form of 思しき in the written language, but it is often 思しい in more spoken language contexts. The form 思しい is not really common, but it does show up periodically in literature. 思わしい doesn't have this restriction, but it’s mainly going to be in the negative form.
Using 思しい to mean こうありたいと思われる is not used anymore and is only found in older/classical texts. It is listed in dictionaries because of its historical existence. This meaning, though, has been taken over by 思わしくない. In 国語 tests, such questions regarding this usage come up, and you would not get points for choosing 思しい. This is because current language usage does not include this definition. Also, 望まれる or the definition こうありたいと思われる may be more common overall.
As for the meaning of showing resemblance, many prescriptive people say that 思しい should be the only one to mean this. Although overall it is what most often appears rather than 思わしい, many speakers use this word to show resemblance. This is reflected in the dictionary definitions above. Thus, this is when textbook definitions and actual usage in society don't match up.
Now it’s time for example sentences to get a sense of the somewhat literary/formal/書き言葉的 style that these phrases are used in and what sorts of phrases fit well with them.
To lose sight of a person thought to be a criminal.
2. 一見、画家と思しき人 （Uncommon usage)
A person who in one look seems to be an artist
I also come across people in my town who appear to be assistant English teachers.
Two days ago when I went to Asakusa, there were a lot of people walking around who looked to be foreigners in a group vacation.
I don't come across foreign-looking individuals in this town.
A woman who appears to be on the run
If you found something that appears to be a hazardous material, immediately contact the police.
The economic has been bad recently, and the number of people who appear to be foreign workers has increased.
There are letters in this letter that appear to be coded.
10a. 線路と車輪から出ると思わしい音 ??
A sound resembling what comes from between the tracks and the car wheels (of a train)
Sentence Note: Because 思わしい is used 90% with the negative, this sentence may sound strange to many Japanese speakers. 10b is a more natural version of 10a that means the same thing, but 10c is a more practical sentence which just lacks the sense of "resembling".
The disease's condition is not satisfactory.
The Olympics' results were not satisfactory.
My wrist's condition was just not satisfactory, you know.
I haven't been so well recently.
I even glimpse animals that appear to be endangered species here and there.
Fossils that resemble chordates
It's said that creatures resembling the ancestors of mankind appeared on the earth 20 million years ago.
Fossils of human species thought to be those of man's ancestors are continuously being found all across Africa.
19. 寝起きと思しく乱れた髪 ? （書き言葉）
Missed up hair like one has woken up from bed
Grammar Note: Using 思しい in the 連用形 as 思しく is rather rare, but it is still possible.
20. テストの結果が｛〇 思わしくなかった・△/X 思しくなかった｝。
My test results were not satisfactory.
21. 犯人と思しい人相の人 （やや書き言葉的）
A person with the looks of a criminal
From the 広辞苑第5版の397ページ
22. 観光客と思しい人たちが結構乗っています。 （Not conversational but perhaps report of some kind) There are quite a few people riding who appear to be tourists.
The following are most certainly possible in the spoken language. After all, と思しき is still occasionally used in the spoken language. It's just not used a whole lot.
I say a person that looked like A-san at the train station.
Something what appears to be dog food has fallen.
I saw a person that appeared to be a criminal.