Whereas the first sentence is completely fine:
Es ist derselbe Verein, der sich auch heute nochauch heute noch um die Tiere kümmert.
your second sentence
*Es ist derselbe Verein, der sich bis zum heute nochbis zum heute noch um die Tiere kümmert
is prima vista not a well-formed sentence in terms of standard German. It is not an established way of speaking to combine heute with bis zum. Heute goes simply with bis, so bis heute is correct.
Accepted and regularly used expressions are
bis heute
bis zum heutigen Tage
Simple reason: zum is derived from zu dem, which is zu plus definite article, and when there is a definite article, there has to be a noun. Heute is not a noun (usually).
This is the core of the answer. Now, for some intricacies:
I said "prima vista" because you can - theoretically - argue that in a certain way of thinking your second sentence is acceptable anyway. It is a little bit a form of twisted thinking, though. And you would have to write the heute as Heute with a capital H.
Es ist derselbe Verein, der sich bis zum Heute noch um die Tiere kümmert
In this sentence, the Heute is used as a noun, and therefore you can say bis zum Heute. However, this is not how anybody would usually speak or write. It is more like a hypothetical sentence constructed in analogy with (more accepted, more well-formed) sentences such as:
Wer das Gestern nicht kennt, verirrt sich im Heute.
Im Gestern und im Heute entscheidet sich das Morgen.
As you may conclude from the broad-brush philosophy there, using heute, gestern, morgen as nouns is a little bit manieristic and would be met only in higher registers of speech.
Oh, and in order to answer your actual question: No, there is no difference of meaning between your sentences. The second sentence would usually be considered as ill-formed, but the meaning of it would be understood quite so as in the first (well-formed) sentence.
Uhh... and one thing more: if you juxtapose "noch heute" and "bis heute", and if you interpret things in a very subtle way, you may argue that
noch heute
would tend to mean "started earlier, continues until today, and will continue in the future (or at least there is no end in sight)", whereas
bis heute
could be understood (but not necessarily) as "started earlier, continued until today, and that's it, no continuation into the future". But this is a little bit of a muddy terrain here. You can also understand bis heute as continuing into the future anyway.
That's of course more about the meaning and usage of noch and bis.