"I used my dictionary a lot."
Translation:Mi multe uzis mian vortaron.
In this context it's similar. But you can say 'I eat a lot' and 'I eat often' and those have different meanings. In this context I can use my dictionary a lot when I do use it, which is not frequently. In that case saying I use it a lot will be true but saying I use it often will not be true.
"I very (strongly) use my…"
Tre is used with "multe" to make "very much" but by itself it just emphasizes the following word. ex: Mi tre laboremis = I was very diligent. Or mi tre ŝatas purpurajn anisidojn = I like purple ducklings a lot (as opposed to: mi ŝatas tre purpurajn…). But uzi, being transitive or something, needs another word besides the tre.
I do hope that this helps.