For instance, I hear "professeur" and "fleur" with the "eu" sound, not "œ".
TL;DR: "œ" and "eu" aren't the same, but "œu" and "eu" do make the same sounds. Exactly which sound depends on context, but the behaviour mostly overlaps between the two of them.
In Parisian French "professeur" and "fleur" both end with [œʁ], using the vowel [œ] as in "œuf".
R-colouring likely comes into play. This is when an "r" modifies the vowel before it. Technically, as Wikipedia states, there's an actual change of vowel only in rare contexts — apparently you should only hear it in Québec French and then only in borrowings like "hamburger". But it would be fair to say that "r" mildly affects the quality of any vowel before it. The combination [œr] in particular can sound very odd to English ears, even though "œuf" doesn't raise any flags.
When going from spelling to pronunciation, always keep a few terms in mind: graphemes (letters/smallest indivisible combination of letters), phonemes (underlying sound), and phones (the different ways a single phoneme actually sounds in a particular context).
It requires a lot of analysis to tell which phoneme a particular letter combination is linked to — and even that would have to be qualified with "most of the time", because spelling isn't consistent in French. (Even if French is much more consistent than English!)
I'd hoped to just copy a previous analysis, e.g. on the Wikipedia French phonology article... but since I don't see this adequately treated there, here's my best shot after testing out some dictionary entries and trying to recall my linguistics education. :p
First of all, the two combinations we're comparing are "eu" and "œu". That's because "œ" behaves a lot like "e". It came into the language as an attempt to spell the Greek diphthong [oi̯] (similar to "boy"). However, its pronunciation "was reduced to a simple vowel ([e]) in late Latin".
Being an attempt to render another language's sound, "œ" unsurprisingly has some inconsistent behaviour depending on which combinations it appears in — but then "e" on its own is also inconsistent. :) Luckily, we can reasonably hope that "œu" behaves similarly to "eu".
Mostly, open syllables (syllables that end in a vowel) are realized as [ø]:
pneu peu feu ceux yeux
And closed syllables (syllables that end in a pronounced consonant) are realized as [œ]:
arroseur peur fleur fleuve
œuvre sœur cœur mœurs mœuf œuf
However, there are some exceptions that trouble this analysis:
arroseuse émeute ([ø] despite being closed syllables)
Particularly interesting is this one, where the phonetic context is the exact same:
jeune ([œ], as expected) vs. jeûne ([ø])
I haven't been able to find such exceptions with "œu", only with "eu". So that is probably a good hint that they can diverge sometimes.
The French phonology article also refers to "longer" syllables closed by a few specific consonants that are all voiced, which affect some similar vowel pairs in French. (The reason phones like [ø] and [œ] are linked to one phoneme is that they are very similar in pronunciation, sometimes differing only by whether the tongue is a little higher or farther forward.)
That might help to explain [ø/œ], but some of this data would still be hard to account for. For example, this group couldn't include both "arroseuse" ("long" syllable) and "émeute" ("short" syllable), nor would it explain "jeune" vs. "jeûne". To get a really robust analysis, we'd need to dig up more data.
Also, "œil" also has [œ], suggesting that "œ" followed by any vowel, not just "u", is pronounced [œ].
When not followed by another vowel, "œ" is often pronounced [e] as it was in late Latin, particularly in technical terminology (which often enters a language late and ends up using alternative phonology that hasn't seen centuries upon centuries of everyday use). For example:
cœliaque cœlacanthe fœtus œsophage
There are always exceptions, such as the loanword "Gœthe" (with [ø], as pronounced in German).
Remember that this is just a provisional analysis... if someone takes the time to find exceptions to base a rule on, or to find an article detailing this in a clearer way, I welcome the insight. :)