I see some mobile operators mentioned LTE and some are mentioning LTE with band 4. What is the difference between those two ?
LTE is a standard for building a mobile network. It lists the available bands on which to operate on and Band 4 is one of those bands. See https://en.wikipedia.org/wiki/E-UTRA#Frequency_bands_and_channel_bandwidths for a full list.
If someone mentions only "LTE" than the band is not specified but you can look up which bands are in used by a specific operator in a given country on https://en.wikipedia.org/wiki/List_of_LTE_networks. For example, where I live (Sweden), that would be band 3, 7, 8 and 20 for my operator but no-one ever refers to it as "LTE with band 3,7,8 and 20", it's just called "4G" or "LTE".
There could be a few different reasons for specifically mentioning the support of a band. If it is from a representative of the mobile operator it is likely they perceive it as a selling point. Lower frequencies have lower attenuation and therefore propagate further enabling better coverage. Higher frequencies are usually handed out in larger blocks so to enable higher throughput. Note how 20MHz of 900MHz spectra will enable the same speed as a 20MHz block in the 2600MHz band.
Another reason, however a bit unlikely to be pitched by a operator representative, is that certain bands are less commonly supported on handsets, so you might want to check that the device you intend to use support the bands that the operator is using.