In Mark 15:43, the Greek text states,
ἦλθεν Ἰωσὴφ ὁ ἀπὸ Ἁριμαθαίας εὐσχήμων βουλευτής ὃς καὶ αὐτὸς ἦν προσδεχόμενος τὴν βασιλείαν τοῦ θεοῦ τολμήσας εἰσῆλθεν πρὸς Πιλᾶτον καὶ ᾐτήσατο τὸ σῶμα τοῦ Ἰησοῦ (TR, 1550)
which is translated as,
Joseph, who is from Arimathaia, an honorable counselor, who himself also was awaiting the kingdom of God, came [and] entered boldly unto Pilate and «ᾐτήσατο» Jesus' body.
In Matt. 27:57-58, the Greek text states,
57 Ὀψίας δὲ γενομένης ἦλθεν ἄνθρωπος πλούσιος ἀπὸ Ἁριμαθαίας, τοὔνομα Ἰωσήφ, ὃς καὶ αὐτὸς ἐμαθήτευσεν τῷ Ἰησοῦ 58 οὗτος προσελθὼν τῷ Πιλάτῳ ᾐτήσατο τὸ σῶμα τοῦ Ἰησοῦ τότε ὁ Πιλᾶτος ἐκέλευσεν ἀποδοθῆναι τὸ σῶμα (TR, 1550)
which is translated as,
57 And after it was evening, there came a rich man from Arimathaia, by the name Joseph, who himself also was Jesus' disciple. 58 After this [man] came unto Pilate, he «ᾐτήσατο» Jesus' body. Then Pilate commanded the body to be delivered.
Both verses have the verb ᾐτήσατο, a verb conjugated from the lemma αἰτέω.1 According to Joseph Henry Thayer (note the bold-faced text),
αἰτέω, -ῶ; fut. αἰτήσω, 1 aor. ᾔτησα; pf. ᾔτηκα; Mid., pres. αἰτοῦμαι; impf. ᾐτούμην; fut. αἰτήσομαι; 1 aor. ᾐτησάμην; [fr. Hom. down]; to ask mid. to ask for one’s self, request for one’s self; absol.: Jas. 1:6; Mt. 7:7; mid., Jas. 4:3; Jn. 16:26; Mk. 15:8; αἰτεῖσθαί τι, Jn. 15:7; Mt. 14:7; Mk. 6:24; 10:38; 11:24; 15:43; 1 Jn. 5:14 sq.; Lk. 23:52; Acts 25:3, 15, etc.; αἰτεῖν with acc. of the pers. to whom the request is made: Mt. 5:42; 6:8; Lk. 6:30; αἰτεῖσθαι with acc. of the pers. asked for—whether to be released, Mt. 27:20; Mk. 15:6 [here T WH Tr mrg. παραιτ. q. v.]; Lk. 23:25; or bestowed as a gift, Acts 13:21; αἰτεῖν τι ἀπό τινος, Mt. 20:20 L Tr txt. WH txt.; [Lk. 12:20 Tr WH]; 1 Jn. 5:15 L T Tr WH; (so αἰτεῖσθαι in Plut. Galb. 20) [cf. B. 149 (130)]; τὶ παρἀ τινος, Acts 3:2; Mt. 20:20 R G T Tr mrg. WH mrg.; Jas. 1:5; 1 Jn. 5:15 R G; foll. by the inf., Jn. 4:9; mid., Acts 9:2; [αἰτεῖν τι ἐν τ. ὀνόματι Χριστοῦ, Jn. 14:13; 16:24 (see ὄνομα, 2 e.); τὶ ἐν τῇ προσευχῇ, Mt. 21:22]; αἰτεῖν τινά τι, Mt. 7:9; Lk. 11:11; Mk. 6:22; Jn. [14:14 T but L WH Tr mrg. br.]; 16:23; ὑπέρ τινος foll. by ἵνα. Col. 1:9 [cf. B. 237 (204)]; αἰτεῖσθαι with the acc. and inf., Lk. 23:23; Acts 3:14; with inf. only. Acts 7:46 (ᾐτήσατο εὑρεῖν he asked that he himself might find; others wrongly translate ᾐτήσατο desired); Eph. 3:13. With the idea of demanding prominent: αἰτεῖν τι, Lk. 1:63; 1 Co. 1:22; τινά τι, Lk. 12:48; 1 Pet. 3:15.
[The constructions of this word in the Greek Bible, the Apost. Fathers, etc., are exhibited in detail by Prof. Ezra Abbot in the No. Am. Rev. for Jan. 1872, p. 182 sq. He there shows also (in opposition to Trench, § xl., and others) that it is not “the constant word for the seeking of the inferior from the superior,” and so differing from ἐρωτάω, which has been assumed to imply ‘a certain equality or familiarity between the parties’; that the distinction between the words does not turn upon the relative dignity of the person asking and the person asked; but that αἰτέω signifies to ask for something to be given not done, giving prominence to the thing asked for rather than the person, and hence is rarely used in exhortation. Ἐρωτάω, on the other hand, is to request a person to do (rarely to give) something; referring more directly to the person, it is naturally used in exhortation, etc. The views of Trench are also rejected by Cremer, 4te Aufl. s. v. The latter distinguishes αἰτέω from similar words as follows: “αἰτέω denotes the request of the will, ἐπιθυμέω that of the sensibilities, δέομαι the asking of need, while ἐρωτάω marks the form of the request, as does εὔχεσθαι also, which in classic Greek is the proper expression for a request directed to the gods and embodying itself in prayer.” Ἐρωτάω, αἰτέω and δέομαι are also compared briefly by Green, Critical Notes, etc. (on Jn. 14:13, 16), who concludes of ἐρωτάω “it cannot serve to indicate directly any peculiar position, absolute or relative, of the agent. The use of the word may, therefore, be viewed as having relation to the manner and cast of the request, namely, when carrying a certain freedom of aim and bearing; a thing inseparable from the act of direct interrogation”; cf. further Schmidt ch. 7. Comp.: ἀπ-, ἐξ-, ἐπ-, παρ·(-μαι), προσ-αιτέω.]
In summary, the same verb ᾐτήσατο occurs in both verses, but Mark adds that Joseph of Arimathaia "entered boldly unto Pilate" (τολμήσας εἰσῆλθεν πρὸς Πιλᾶτον), not fearing any repurcussions. There's no basis for suggesting that one gospel describes Joseph as begging while the other describes him as boldly requesting. The same verb expresses the same action in both, albeit Mark's account is slightly more detailed.
1 3rd person, singular number, aorist tense, indicative mood, middle voice
Thayer, Joseph Henry. A Greek-English Lexicon of the New Testament: Being Grimm Wilke’s Clavis Novi Testamenti. New York: American Book, 1889.