What is the difference between main lug and breaker box

What is the size of feeder wire most commonly used from the meter to the breaker box

Okay, you have two questions here.

The cabinet for a service panel more than likely is the same.

The difference between a MLO "Main Lug Only" and a Main Breaker Box is the primary protection for the panel board.

A service Panel Board with a Main Breaker is exactly that. It will have a breaker equal to the incoming service. If there is a problem, the main breaker will trip and prevent damage caused by the overcurrent.

A MLO service panel board however does not have the protection of the main breaker and therefore it needs to conform to the NEC code.

It can be protected by no more than six breakers or a "Main Disconnecting Means"

As to the cable size, that is totally dependent on the size of the service amperage and whether you are speaking of a sub panel or a main panel.

For example, if you are feeding the Main Service Panel, then you would use Table 310.15 (B)(6).

If you are feeding a sub-panel, then you need to use Table 310.16.

You also need to select the correct cable type, based on insulation type, whether the cable will be above ground (service drop) or below ground (service lateral).

Distance is a factor also.

Is not a simple process. Also, you need an electrician to do a load calculation to make sure that you have the correct service amperage ordered.

In short, I suggest you contact several licensed electricians to get bids on doing the work and in particular the calculated load vs. connected load.