What is the difference whether one wears a tallit or not during תפילה (prayer/worship, sacred service)?
The wearing of a tallith messuyesseth during tefillah is very important. However, it does not prevent one from praying should the time for prayer arrive and he find himself without a tallith - whether gadhol or qatan.
The Rambam in the Mishneh Thorah (Hilkhoth Tefillah 4:1ff) lists five things which prevent one from praying, even if the time for prayer has arrived. The wearing of a tallith is not among them. Additionally, the Rambam (Hilkhoth Tefillah 5:1) lists eight things which one should be careful to fulfill during prayer, but do not prevent one from praying should they be left unfulfilled. Again, the wearing of a tallith is not among them.
In Hilkhoth Ssissith 3:12-13, the Rambam expresses the importance of the misswah of ssissith, and especially during prayer. He says:
uvsho`âth ha-tafilloh ssorikh le-hizoher bi-yôther, genoi godhôl hu le-thâlmidh hâkhomim shayyithpâllâl wa-hu eino oTuf.
"And during prayer, one must be even more careful [to be wearing ssissith, i.e. more than at other times of the day], for it is a great disgrace for a talmidh hakhamim [a Torah scholar] to pray and not be wrapped [in a tallith messuyesseth].
So, while it is important for each Jewish man to both acquire and wear a four-cornered garment with ssissith, it does not prevent one from praying (or reciting Qiryath Shema`).
And, being that the Rambam mentions it especially regarding Torah scholars, it indicates that he feels that it is imperative for them to wear a tallith during prayer because of their connection to the Torah itself (cf. Hilkhoth De`oth 5:1ff) - although it still would not prevent them from fulfilling their obligation of prayer.
As for the commonly cited idea that not wearing ssissith during Qiryath Shema` causes one to be me'idh sheqer be-assmo ("to testify falsely concerning himself"), this is an idea that stems from the Zohar - it is mentioned nowhere in the corpus of Hazal (i.e. Talmudh Bavli, Talmudh Yerushalmi, Mishnah, Tosefta, Mekhiltoth, Sifra, Sifrei - cf. Rambam, Haqdamah le-Mishneh Thorah 24).
In the Arokh HaShulhan, Hilkhoth Ssissith Wa-'Attifatho 8:1, it discusses the apparent contradiction between the assertion of the Zohar and the halakhah in this matter. There he states that this statement of the Zohar is referring to someone who recites Qiryath Shema` while wearing a four-cornered garment which does not have ssissith tied onto it - dawqah. [See there for a full discussion of this subject]
As far as their prayer is concerned, there is no difference between one who wears a tallith messuyesseth and one who does not. However, the one who is able to wear ssissith during prayer and fails to do so has been lax in his commitment to wear them during their prayers. Of course, if one finds himself at the time of prayer but without a tallith messuyesseth that he can put on, he should certainly not delay his prayers or refrain from praying on account of such circumstance.
Hope that this helps. Kol tuv.
ועל ידי מצות ציצית, תנצל נפשי רוחי ונשמתי ותפלתי מן החיצונים
and through the mitzva of tzitzis may my soul, spirit and prayer be protected from (harmful) external influences
So it seems that the tallit somehow protects the thoughts of the person praying.
O Ch 24 (1) says
that a person should wear tzitzis (tallis koton) all day and if he does not he should at least wear a tallis godol when he prays.
MB  says that
someone who says Shema without tzitzis is as if he testifies falsely for he says the parsha of tzitzis and does not fulfil it.
He quotes the Chayei Adam who says that
people on a journey who daven without a tallis godol do not do well because the majority of the tallisos ketanim are not made according to the law in all its particulars.
So another reason to wear a tallis godol is avoid saying the parsha of tzitzis and not fulfilling it.