In IPA chart, there is a column named "palatal consonants", including consonants as ɲ, c, ɟ, ç, ʝ, ʎ for example.
There is also a 'palatalization sign': ʲ, which can be applied to all consonants, used, for example, in Slavic languages.
What are the differences between of consonants of the palatal columns, and there palatalized doubles nʲ, tʲ, dʲ, ʃʲ, ʒʲ, lʲ? Or it is just two writtens for the same sound?
Theoretically, there is a difference in most cases.
In IPA, the raised j symbol <ʲ>, represents "palatalization," or a "palatal secondary articulation." The concept of a "secondary articulation" is itself somewhat vague. A palatal secondary articulation might occur simultaneously, slightly before, or slightly after the primary articulation of a consonant, or even be realized mostly as a modification of an adjacent vowel. There are general patterns that naturally occur, but the extent to which they occur varies between languages. So in some languages, the phoneme /nʲ/ might be automatically realized before a vowel with a semivowel offglide (/nʲV/ = [nʲjV]). However, in other languages, such as Russian, sequences of /CʲV/, /CV/, /CʲjV/, and /CjV/ are all phonologically distinct. (Russian non-palatalized consonants are sometimes said to have velar secondary articulation, represented by the sign <ˠ>, so that "/CjV/" in the previous sentence could also be written as /CˠjV/ I guess.)
The symbols /ɲ, c, ɟ, ç, ʝ, ʎ/ represent a consonant with a single place of articulation that is palatal (palatal "primary" articulation).
The symbols /nʲ, tʲ, dʲ, ʃʲ, ʒʲ, lʲ/ represent a consonant with a primary place of articulation that is coronal (there are various more specific classifications of coronal consonants such as dental, alveolar, laminal), and a secondary place of articulation that is palatal.
In practice, however, this is an area where people often use symbols approximately. There are a range of phonetic variants that can be symbolized in different ways. Phomemically, many languages do not distinguish between pure palatal consonants and palatalized coronal consonants (e.g. /nʲ/ vs. /ɲ/), but some do. However, if I'm understanding a comment by User6726 correctly, there may be theoretical reasons to suppose that no language has a contrast between pure palatal consonants and palatalized velar consonants (e.g. there is no language with distinct /ŋʲ/ and /ɲ/, apparently).
Nonetheless, the symbols for palatalized velars like <kʲ> <gʲ> are used in a fair number of transcription systems, perhaps to indicate that in the relevant language these sounds are phonologically related to plain or back velars.
[ç, ʝ] are canonically "pure" palatal sounds. These are pretty clearly different phonetically from [ʃʲ, ʒʲ], which are alveolo-palatal (and equivalent to [ɕ, ʑ]).
Japanese even has an arguably phonemic distinction between these sounds: ひゃ is typically transcribed as [ça] while しゃ is typically transcribed as [ɕa]. I have read that these sounds are merged or near-merged in some accents of Japanese, but the standard form of Japanese recorded in textbooks for foreign learners keeps them distinct.
I say "arguably" because there is a tradition of analyzing these examples as phonemically being /hja/ and /sja/ or respectively, which makes the phonological contrast an opposition between the segments /h/ and /s/. This is certainly more-or-less valid as a historical and morpho-phonological analysis; however, I'm less sure of its validity as a synchronic description of the phonological representation for contemporary Japanese speakers. A paper I found that touches on the issue of phonological representation of "yōon" syllables in Japanese: "Acquisition of Yo-on (Japanese contracted sounds) in L1 and L2 phonology", Chiharu TSURUTANI, describes the sounds as palatalized versions of /h/ and /s/ rather than consonant clusters; this could imply the phonological transcriptions /hʲa/ and /sʲa/ (although the version of the paper that I can see seems to use /hj/ and /sj/).
Again, [c, ɟ] are defined as pure palatal plosives. These phones are not very common (unless you consider fronted allophones of phonemically velar /k, g/ in some languages to be front enough to qualify; e.g. /k/ in French "qui"). A place of articulation in between velar and palatal can be called "post-palatal" or "pre-velar", and represented by [c̠, ɟ̠] or [k̟, ɡ̟].
[tʲ, dʲ] may represent alveolo-palatal plosives [t̠ʲ d̠ʲ], which apparently exist in Korean. According to Wikipedia's page on Russian phonology, Russian /tʲ, dʲ/ are phonetically "laminal alveolar" and affricated; Wikipedia uses the transcriptions [t̻ʲsʲ, d̻ʲzʲ].
There are some languages that make a two-way distinction between palatalized coronal plosives and palatal plosives, such as Irish, which has /tʲ/ as the palatalized counterpart of /t/ and /c/ as the palatal counterpart of /k/.
As I mentioned earlier, in many languages velar phonemes are allophonically fronted when adjacent to front vowels or semivowels. But some languages have a phonemic distinction between fronted velars and other types of velars.
There is a fairly clear theoretical distinction between the IPA phones [ɲ, ʎ], which are canonically defined as "pure" palatal sounds, and [nʲ, lʲ], which indicate alveolar or dental primary articulation with secondary palatal articulation (the most common variant is the "alveolo-palatal" articulation [n̠ʲ, l̠ʲ], which Wikipedia says may also be represented as [ɲ̟, ʎ̟]). Another possible sound in this region is "post-palatal" or "pre-velar" which is between the canonical "palatal" and "velar" places of articulation; a post-palatal nasal could be narrowly transcribed as [ŋ̟].
In broad phonetic transcription or phonological transcription, however, /ɲ, ʎ/ are often used to represent anything from an aveolo-palatal, to a true palatal, to a post-palatal consonant. Most languages do not have a phonological contrast between these. There are some languages that make a two-way distinction, such as Irish, which has /nʲ/ as the palalized counterpart of /n/ and /ɲ/ as the palatal counterpart of /ŋ/. Apparently, some dialects of Irish even make a three-way distinction in this area due to a phonemic distinction between "fortis" and "lenis" forms of /nʲ/; I don't know enough to understand how this distinction is realized phonetically in the various dialects that still maintain it.
Spanish /ɲ/ is, according to Wikipedia, something like an alveolo-palatal, although as michau pointed out in a comment it is kept distinct from the alveolar nasal /n/ even before /i/ (where /n/ is somewhat palatalized), and before vowels other than /i/, /ɲ/ is distinguished from the sequence /ni̯/.
