This answer details a bit of history of the development of「于」and「於」, and addresses why they overlap. It is purely meant to be a complement to @user3306356's answer, which has responded to the question directly.
In the Modern Chinese language shared across all Chinese-using regions,「于」and「於」largely represents the same overlapping preposition/adposition function. For this function only*, any proposed "rules" specifying the difference in usage between the two are very unlikely to be accepted across all Chinese-speaking regions, mainly complicated by the fact that PRC has made「於」obsolete and Taiwan has a habit of using「於」and not「于」. This is also why you wouldn't find「于」on all but the most thorough Taiwan-Chinese input methods.
「于」and「於」are both phonetic loans; they both originally represented something else.
Glyph origin of 于
「于」was originally a picture of a wind instrument:
It was later simplified to
then further simplified to「于」. The original sense of the wind instrument is now written「竽」after「于」became extensively used for its grammatical function meaning and had lost its original meaning.
Glyph origin of 於
「於」and「烏」were originally the same character, but have been graphically differentiated. For reference, their development looked something like
There was absolutely no difference between the characters「於」and「烏」before the Qin dynasty, and「烏/於」was used for its current grammatical function long before they became different characters.
The historical usage of「于」and「於」goes in accordance to something like the following timeline:
Western Zhou
Late Western Zhou to end of the Spring and Autumn period:
See also 從“于 / 於”用法上的變化看複合詞“於是”的產生.
*There are differences between「于」and「於」when they're not used in the grammatical function sense, which this answer touches upon but doesn't explain thoroughly - detailed dictionaries will provide a more comprehensive explanation.
Check out the two entries in A Student's Dictionary of Classical and Medieval Chinese for when to use 于 and not 於:
於
MC 'jo
1 (GP) typically occurs after a VB or VB-phrase and before a noun or noun-phrase, indicating any indirect or locative relation: in, at, from, to, on, with regard to, vis-à-vis, etc.; e.g. 王坐於堂上 wáng zuò yú tángshàng, The king was sitting in the hall; 千里之行始於足下 qiānlǐzhi xíng shǐ yú zúxià, A journey of a thousand li starts from under one’s foot; 敏於事而慎於言 mǐn yú shì ér shèn yú yán, diligent vis-à-vis affairs and careful vis-à-vis speech; cog. yú 于 in, at, from, to, on, with regard to, etc.
N.B. Fom the Warring States period on 于 yú and 於 yú are generally interchangeable, save for stylistic preferences; prior to the Warring States period 於 yú is rare, and the two are not equivalent to each other.
a) (GP) after a VB or VB-phrase and before a noun or noun-phrase, indicating comparative sense: than; e.g. 民少於今日 mín shǎo yú jīnrì, The citizens were fewer in number than [those of] the present day; 青於藍 qīng yú lán, bluer than indigo.
2 (Budd.) (GP) in Budd. texts often simply a marker preceding the obj, with no translatable meaning.
于
MC hju
1 typically occurs after a VB or VB-phrase and before a noun or noun-phrase, indicating any indirect or locative relation: in, at, from, to, on, with regard to, vis-à-vis, etc.; e.g. 入于幽谷 rù yú yōugǔ, entering into a dark valley; 君告于天子 jūn gào yú tiānzǐ, The lord reported it to the Son of Heaven; cog. yú 於 in, at, from, to, on, with regard to, etc. N.B. From the Warring States period on 于 yú and 於 yú are generally interchangeable, save for stylistic preferences; prior to the Warring States period they are not equivalent to each other.
a) preceding a time-word (and often after zhì 至): arriving at, up to, at [the time in question]; e.g. 而至于今 érzhì yú jīn, and still it continues up to the present; 于時以為蜀得其龍 yú shí yǐwéi shǔ dé qí lóng, by that time they figured that Shu had gotten its dragon; 二世三世至于萬世 èrshì sānshì zhì yú wànshì, for two generations, three generations, up to a myriad of generations.
b) after a VB or VB-phrase and before a noun or noun-phrase, indicating comparative sense: than; e.g. 多于前功 duō yú qiángōng, more than previous merit; 異于今人 yì yú jīnrén, more unusual than people of the present day.
2 pre-classical VB prefix of uncertain meaning, possibly an inchoative sense or rhetorical emphasis; e.g. 黃鳥于飛 huángniǎo yú fēi, The yellow birds are taking to flight; 之子于歸 zhīzǐ yú guī, That fellow is on his way home.
3 (bn.) 于闐 yútián (MC hju-den), Khotan, oasis city on the southern Silk Road in the Tarim Basin, the main source of nephrite imported to China.
4 arch-shaped rim or sound-bow of a chime-bell’s mouth.
You can see from the reconstructions that 於 and 于 didn't even share a common pronunciation.
These are classical and medieval usages though. In modern Chinese the difference is less obvious.
The usage of 致力于 and 歸于 on that site you linked above is somewhat baffling. 两岸 has entries for: 致力於 https://www.moedict.tw/~%E8%87%B4%E5%8A%9B%E6%96%BC and 歸於 https://www.moedict.tw/~%E6%AD%B8%E6%96%BC Taiwan's Ministry of Education certainly doesn't see any reason to use 于 over 於.
MoE does make some distinctions as well:
介詞：
➊ 引進時間、地點、範圍等，相當於「在」。例 「生于1949年」、「發源于黃河」、「耿耿于懷」。
➋ 引進方面、原因、目的。例 「勇于承擔」、「死于車禍」、「急于做到」。
➌ 引進方向、目標。例 「言歸于好」、「嫁禍于他人」、「獻身于教育」。
➍ 引進對象，相當於「向」、「給」、「對」。例 「忠于國家」、「有益于人民」、「無濟于事」。
➎ 引進比較的對象。例 「重于泰山」、「輕于鴻毛」、「高于樓臺」。
➏ 引進行為的主動者。例 「死于槍手」、「限于篇幅，暫不刊登」、「國家隊竟敗于默默無聞的隊伍」。
➐ 引進來源、起點，相當於「從」、「自」。例 「青出于藍」、「佛教發源于印度」。 ∥也作「於」（陸 今不用）。
助詞。用在句首或句中，無意義。例 「鳳凰于飛」、「之子于歸」。
姓。
and
简于
介詞：
引進地點、範圍、時間等，相當於「在」。例 「飯店位於機場對面」、「於無意中發現這個祕密」、「生於70年代」。
引進對象，相當於「向」、「給」、「對」。例 「問道於盲」、「於事無補」、「己所不欲，勿施於人」。
引進來源、起點，相當於「從」、「自」。例 「青出於藍」、「佛教發源於印度」。
引進行為的主動者。例 「地主隊敗於客隊」、「見笑於大方之家」、「限於篇幅，忍痛割愛」。
引進方向、目標。例 「天氣逐漸趨於穩定」、「致力於學術研究」。
引進方面、原因、目的。例 「勇於負責」、「苦於沒有機會」、「執著於實現理想」。
引進比較的對象。例 「重於泰山」、「苛政猛於虎」、「成長率高於預期」。 ∥也作「于」。
Most of the definitions are more or less the same and it's quite hairy in general.
