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.
Fricatives: [ç, ʝ] vs. [ʃʲ, ʒʲ]
[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/).
Plosives: [c, ɟ] vs. [tʲ, dʲ]
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.
Sonorants/resonants: [ɲ, ʎ] vs. [nʲ, lʲ]
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/.