My dictionary shows kara twice:
kara: from kara: since, because
Is there a difference in the way these two are written or is there just one meaning?
I'm going take a different tack on this from everyone else.
If you go back to earlier Indo-European languages such as Latin, but more so like Sanskrit, you see a very similar use of the ablative case to express both the idea of "from" and "because". There are still vestiges of this in English when we say something along the lines of
From what I've seen,....
which can easily be rephrased as
Because of what I've seen,....
In Japanese, the difference in the meanings should be clear from context. If から follows a noun, the most likely meaning is "from". If から follows a verb, then it's most likely going to mean "because" unless it's follow the "te" form of the verb, in which case, から will mean "after".
Generally, not only in this case, determining if multiple definitions of a word form are homonymous (i.e. being distinct concepts) or polysemous (i.e. being a single concept) is a hard problem in theoretical linguistics.
It's safe to say the two definitions of kara you cited are usually covered under the same headword in Japanese-Japanese dictionaries, because they are etymologically identical, and have relatively clear semantic similarity, but grammatically not equal, being postposition and conjunction, respectively.
The effort to delimit exact borderlines between meanings tends to be unproductive and opinion-based. For example:
The U.S. should elect a woman as president "as soon as possible,"... (source)
Are the three instances of as same meaning? At least, they all belong separate parts of speech, so very different in function. Some English dictionaries list them as as1, as2..., but Japanese dictionaries don't like this way.
You cannot rely on translation, either. Are "sleeping" dream and "fancying" dream same thing? They share the same word in many languages, including English and Japanese, but are separate ideas in many other languages, including Finnish and Russian.
A dictionary for use by english speakers learning japanese may list both of those meanings because in english there is a more definitive difference between these two meanings.
から is always spelled the same for these two contexts in japanese. Lets take a look at them both, followed by a tip on what to do with から as you learn.
Take for example the following
"I came from America."
In this sentence, から would be like "from" in English.
The following would be an example of "because" in english.
"Because I understand Japanese, I speak to Japanese people"
Both of these are spelled the same and the nuance of the meaning depends on the context.
I would recommend that you treat から as a sort of a different thing than the two meanings you have been given in english. Try not to treat them seperately.
To a japanese speaking person, there is not really a difference in the same way you would see a difference in english. から is just から and so as you learn to view it that way, it helps your mastery of the language.