Space: less actually (1 TB instead of 2), but only 500 GB were used. And the 2 To were rather expensive, so I'll wait before moving again (or move just the comments-pics or vehicles-pics without moving the whole site). The idea was to have a cheaper server due to Google Ads problems.
Speed problems were rather due to the architecture of the database and its size. That's one of the things that we hope to solve with the new site.
The new server has a difference processor (much newer, but more entry-level), I do not know yet if and how it will impact performance.
It has more RAM (8 GB instead of 4) so that could help with the database slowdowns (more data in cache, faster access).