There are more differences between Slovak and Czech. Just like between Slovak and Polish language, the grammar, vocabulary and accent differ. Many words are common, but they have different form.. it depends. The rest is completely different (just like a war – > vojna in Slovak, válka in Czech). The alphabets are similar, because the Czech type of Latin script had been once used by Slovak intelligence, so it remained. Though both languages have also specific letters (ř, ě, ů in Czech; ä, ô, ŕ, ĺ, ľ, dz, dž in Slovak). If you ask about the main difference, it's the accent – Czech sounds harder than Slovak, Slovak sounds soft. But don't forget, there are Moravian dialects between the Czech and Slovak language. Till the 18th century Moravian dialects were considered to be a 'sibling' of the Slovak language, or more precisely of the western Slovak dialects, rather than to Czech. It has its historical base, 'cause the south Moravian region is called Slovácko (older name Moravské Slovensko /Moravian Slovakia/) and on the other hand our common historical state was called Magna Moravia (it's pity we don't know the name of it in our language, it's the name given by foreigners). Moravians as well as Czechs are our brothers, but Moravians are the closer ones – it's because of more similarities Slovaks and Moravians have in common. Although I consider Moravians as an individual nation (though they have no state, no codified language and Czech politics has problem with it), Moravia is mostly Czechised now.
Zrkadlo, some central Slovak dialects? That is standard Slovak which is based on central Slovak dialect, but that is definitely not "some central Slovak dialects".