This is to signify Tommy Oliver's legacy as a Power Ranger.
Tommy gets his morphing style from his previous ranger forms.Tommy Oliver gets the shape of the morpher and style of morphing [screw driver like thing] from Power Rangers Turbo. He gets his action from White Ranger form from Mighty Morphin Power Rangers
Source Video for the Gif.
Tommy Oliver is the oldest living Power Ranger. He has been in other teams of Power Rangers in his life too.
From Power Rangers Wikia,
Dr. Thomas "Tommy" Oliver is a Power Ranger veteran and considered a legend among the Ranger community. He is often considered to be the greatest Power Ranger of all time, having been part of four Power Ranger teams over the years.
He has been following rangers.
Green Ranger (Evil) - Mighty Morphin Power Rangers. This was a temporary power.
White Ranger - Mighty Morphin Power Rangers.
Zeo Ranger V - Red - Power Rangers Zeo.
Red Turbo Ranger - Power Rangers Turbo.
And finally Black Dino Ranger in Power Rangers Dino Thunder.
In the episode Legacy of Power of Power Rangers Dino Thunder, the other Power Rangers Connor, Ethan and Kira get to know the past life of Tommy Oliver and knows that he was a Power Ranger in multiple teams. They watch his video diary and know the history of Power Rangers too.
It is in the very next episode (Back in Black), Tommy becomes the Black Ranger. So, it is hinted in the episode 4 that there would be some connection with old power Rangers.
The action Tommy does after morphing is taken from his White Ranger action. They look the same.
So, the difference of transformation is inspired from his old power ranger forms.
On a side note, In episode 27, Fighting Spirit, the old power Ranger forms of Tommy Oliver appear in a single episode and had to fight with them to get back his dino gem.