The debate was not about using telescopes, Hevelius was using telescopes since 1647, but about "telescopic sights", micrometers with attached lenses and crosshairs introduced by Hooke, Wren and in France Auzout in 1660-s. It started innocently enough as an exchange of letters in 1665-68 through Oldenburg, a secretary of the Royal Society, and then escalated into a public spat after Hooke's 1674 pamphlet in response to Hevelius's book Machina Coelestis Pars Prior published in 1673 (Cometographia was written before the debate took shape). The context, the substance and the consequences of the debate are described and analyzed in great detail in chapter 4 of Saridakis's 2001 thesis available online. The last episode of the controversy, which reignited in 1685, is covered in the Moxham's video lecture, which focuses on social context.
As early as 1665 Oldenburg communicated Hooke's opinion that "few persons can distinguish an angle less than 1$'$", and that telescopic sights allow to make "instruments of much less bulk, to do ten times more". Hevelius replied in 1667 and asked for full description of telescopic sights fitted to sextants, and for inexpensive lenses of long focal length. But he also made his first arguments against the telescopic sights, and asked for a proof that telescopic sights are more accurate by comparing eight observations of angular distances between stars. Hooke never sent the lenses, despite being prompted to do so by the Royal Society, or took up Hevelius's challenge. The dispute escalated in 1673 after Hevelius gave an expanded version of his arguments in Machina Coelestis without mentioning Hooke.
Hevelius was an old school astronomer in the tradition of Tycho Brahe, he believed in using large instruments, including large telescopes, and "combining measurements from many instruments over a long period" to improve accuracy. In his 1674 pamphlet Animadversions Hooke didn't confine himself to rebutting Hevelius's arguments, but resorted to slights, implying that Hevelius's instruments were no better than Tycho's, and that he wasted his life when he could do so much better using telescopic sights. In this Hooke overestimated telescopic sights of his time, and underestimated Hevelius's instruments and skill. He achieved the accuracy of about 30$''$, whereas Flamsteed using telescopic sights only improved it to 15$''$-20$''$, hardly "ten times". When Halley was dispatched in 1679 to convert Hevelius to the use of telescopic sights his observations with a 2-foot quadrant did not prove superior to Hevelius's, who used 6 or even 9 feet quadrants with open sights. Only years after Hevelius’ death, when Flamsteed completed his new star catalog, did the advantage of telescopic sights become apparent.
The acerbic tone Hooke took, wrapping himself into a mantle of the defender of progress against Hevelius's "attacks", also rubbed many the wrong way including Molyneux and Wallis, who rebuked him publicly for not delivering on the promises of greater precision. Others, like Halley and Flamsteed, were put into an uncomfortable position of agreeing with Hooke on substance but having to distance themselves from his hostile attitude. The Royal Society got involved because such public opposition to telescopic sights from one of the Europe's leading astronomers threatened their general acceptance and use, but favored resolving the issue in a non-confrontational manner, hence the Halley's 1679 mission. According to Saridakis the controversy "entailed such issues as witnessing, etiquette of communication, and the role of the Royal Society as a forum in which individuals could express dissent." Moxham characterizes it as "not just a clash between two notable and notoriously egotistical natural philosophers but a complex set of negotiations between three British learned societies which developed into an open competition for ascendancy."
Hevelius argued that "lenses cannot be maintained in the same position. Consequently, an observer could not acquire the same exact measurements each time he used telescopic sights. On the other hand, with open sights he can repeat the same observation with equal precision each time because he did not have to worry about shifting telescopes caused by their lack of being “firmly fixed.” Besides, lenses could easily break, especially in cold weather. Furthermore, winter weather could fog up or dull lenses when the breath of an observer came into contact with them. As a result, the lenses needed to be frequently removed and cleaned... Cross-hairs (their intersection) tended to conceal small stars so that an observer had difficulty distinguishing them... contributed to inaccurate measurements because of their close proximity to the observer’s eyes (a few inches). Therefore, telescopic sights were less effective than open sights in which the sights themselves were six, or even eight feet from the eyes... proximity between the eye and cross-hairs in telescopic lenses could not ensure an accurate line of collimation."
