You have a decent tallis you wear seven days a week. It’s not old or schmutzy. Should you still try to buy a tallis just for Shabbos? In a word, yes.
Let’s start with the verse in Yeshayahu (58:13):
אם תשיב משבת רגלך עשות חפצך ביום קדשי וקראת לשבת ענג לקדוש ה’ מכבד וכבדתו מעשות דרכיך ממצוא חפצך ודבר דבר
If thou restrain thy foot because of the Sabbath, from pursuing thy business on My holy day;
and call the Sabbath a delight, the holy day of the Lord honorable, and shalt honor it,
not doing thy own ways, nor pursuing they own business nor speaking of vain matters
The Gemara (Shabbos 113a) says that from the word וכבדתו that your Shabbos attire should not be the same as your weekday attire. Based on this Gemara, the Tur writes:
וישתדל שיהיו לו בגדים נאים לשבת דכתיב “וכבדתו” ודרשינן שלא יהא מלבושך של שבת כמלבושך של חול
Pay attention to the first word ישתדל, i.e. one should strive. I once heard Rav Schachter of Yeshiva University say ideally you should have special Shabbos clothes from head to toe: not just a Shabbos shirt, but a Shabbos undershirt, Shabbos socks, Shabbos pajamas, etc.
Apparently this is the meaning of the Mishna Berura 265,2, 5 based on the Arizal. From my own reading of the Mishna Berura I can’t see Rav Schachter’s opinion, since the Mishna Berura says “even the shirt,” which sounds to me like it would come to exclude undershirt and socks. But Rav Schechter was invariably relying on Achronim I haven’t seen.
And now for the key words:
ואם אפשר לו, טוב שיהיה לו גם טלית אחר לשבת
There you have it – “if possible, you should have another tallis for Shabbos.”
Some people have pretty much the same tallis, just less worn. Others may have a fancy atara or ornate Yemenite fringes on their Shabbos tallis. And some people choose to go one size bigger for their Shabbos tallis; during the week, short and more manageable, on Shabbos long and elegant.
I used to wear a black-striped tallis katan with a straight hem during the week, and a white-striped tallis katan with fringes on Shabbos. My current Shabbos tallis has a cotton lining, lending it a more substantial feel, while my weekday tallis does not.