Why can't Naruto go into Bijuu mode like the other Jinchuuriki? Instead, he forms a transparent cloak in the size of the Bijuu, but doesn't get a full form like other Jinchuuriki. Even when fighting Pain , he went up to the 8th tail and got Kurama's real body in a sort of skinned form.
There have been a lot of incomplete answers. Let us try to combine what all we know. This will be spoiler heavy for those who haven't finished the Manga/Anime

TL;DR. When he first used Nine Tales Chakra he COULDN'T go into Bijuu mode. He could once Kurama agreed to work together.


Naruto is the jinchūriki of Kurama's Yang half. Due to his Uzumaki lineage and being Kurama's jinchūriki his entire life, Naruto's chakra has more effectively mixed with the fox's, allowing him to perform several chakra-taxing techniques without feeling fatigued. 
While accessing Kurama's chakra, typically through sheer rage, enhanced Naruto's fighting abilities, the fox's negative influence made Naruto more aggressive and in less control of his actions. This caused him to uncontrollably access Version 1(Tails 1-3 for Naruto) and Version 2 (Tails 4-8 for Naruto) for forms, and needed Kurama's power to be suppressed through various methods to regain control


  Version 1 – or what Jiraiya calls Naruto's "demon fox cloak" (妖狐の衣, yōko no koromo) – forms a dense shroud of chakra around the jinchūriki. The shroud, a translucent red with bubbles of chakra (reminiscent of boiling liquid) forming all along it, provides a degree of physical protection, but can also harm the user if used to a certain extent. The physical changes experienced in the initial transformation become more pronounced in Version 1 forms. 
  
  Version 2 (バージョン2, Bājon Tsū), as described by Sabu, is the tailed beast's chakra being converted into a humanoid shape, granting the jinchūriki an edge in battle without completely releasing the beast itself. Using the jinchūriki's body as a sort of endoskeleton, a dark red, nearly black layer of chakra envelops them. The physical manifestations of the corresponding beast that are vaguely present in Version 1 states become far clearer in Version 2, in a way fully reproducing the beast in miniature: muscle mass increases and beast-specific attributes such as shells and ears become clearly defined.
  

Realising he could not avoid using Kurama's power, Naruto sought to learn how to control it with help from Killer B, the jinchūriki of Gyūki, and unexpectedly, his own mother, Kushina. Upon succeeding and taking most of Kurama's chakra, Naruto attains Nine-Tails Chakra Mode, which greatly increases his speed, strength, and defences. Because he didn't yet have Kurama's cooperation, however, Naruto needed to exercise extreme caution while using the form, as it could cost him his life. He can't do the Bijuu Transformation at this point

After earning Kurama's cooperation, Naruto can enter the Tailed Beast Mode, but the mode, though accurate in size, is unique as he does not as closely resemble the beast as the other jinchūriki do their own. For all intents and purposes, Kurama Mode is a completed version of Nine-Tails Chakra Mode: Tailed Beast Mode without the full transformation, meaning that Naruto is able to use Tailed Beast Balls
 
With the power given to him by Hagoromo, Naruto gained access to the Six Paths Sage Mode (六道仙人モード, Rikudō Sennin Mōdo). In this mode, his eyes become yellow and his pupils take on a cross-like shape — without manifesting the orange pigmentation around his eyes present in Sage Mode. Naruto can also don a new chakra cloak similar to Nine-Tails Chakra Mode, which he can access instantly and maintain much longer than Sage Mode. Naruto's Six Paths Sage Mode cloak changes several times throughout his later years

We see Naruto's Bijuu ability multiple times throughout Boruto movie. He protect the village by stopping the Bijuu Dama kind of attack. Later he and Sasuke combine Kurama mode with Susanoo.


Now to answer the completely DIFFERENT and Duplicate question. From Reading the comments.
- Why do Kurama and Gyuki appear as chakra while other bijuu don&#39;t?
- Another theory is that Naruto's special seal devised by the 4th Hokage causes this. He used the Uzumaki techniques to create the seal. This is why he also appears in a similar Bijuu form as Naruto. Thus, this is a host specific form and NOT Jinchuriki specific. And yeah, Naruto is special.
Reddit for this discussion. Contains Jiraiya's And B's comments on special nature of Naruto's seal

Sources: Naruto Uzumaki Jinchuriki Forms 9-tails Chakra Mode