I'll pick four main differences: poetic form, linguistic form, performance context, and genre.

To begin with, let's stipulate that when we say "Greek lyric", we mean that in a general sense, not the technical sense of "poetry performed to the accompaniment of a lyre". So for the purposes of this post, lyric also includes choral, iambic, and epinician poetry.

• Poetic form. Homeric epic is in a rhythm called dactylic hexameter consisting of a regular march-like rhythm, with an ongoing dum-diddy-dum-diddy riff. Greek lyric poetry comes in a range of many different metres.

A key difference between the two is that hexameter is an isochronic metre, which means that the important thing about each line is that it occupies the same length of time; within that time, different rhythms can be substituted, subject to some constraints. So for example you can regularly substitute dum-dum for a dum-diddy (or in more technical terms, two crotchets in place of a crotchet and two quavers). Lyric metres, by contrast, are isosyllabic, which means that it's the number of syllables in the line that counts: rhythm (or rather what modern western music calls "rhythm") takes a backseat to the syllable-count. This is important because it is believed that Proto-Indo-European poetic forms were generally isosyllabic in nature. The origin of an isochronic rhythm like the hexameter is therefore a bit of a problem, and a subject of ongoing dispute.

• Linguistic form. The isochronic nature of the hexameter seems to have developed out of a system of regular or semi-regular formulaic phrases, which acted as a kind of stockpile that a poet could draw on for improvisatory purposes. This leads to some odd linguistic features in Homeric poetry. In particular, it's often hard to pin down syntactical breaks (where to put full stops and commas), and even word-boundaries can be a bit vague because the semantic units are often formulas rather than the individual words. This formulaic system developed over time (exactly how long a time is disputed) and absorbed features from multiple dialects (again, how this absorption worked and the chronology of the absorption is disputed). Some formulas are extremely ancient (perhaps pre-Mycenaean), others are very late developments: it was a system in constant evolution.

Lyric poetry, by contrast, developed without an extensive formulaic system of that kind, and tends to stick to one dialect at a time. Still, there was seepage from the hexameter formulaic system into lyric language, especially in an intermediate genre called "elegiac" poetry. There are a very few interpreters who would argue that the formulaic system originated in an (Indo-European) lyric context, and that the hexameter developed out of that; but since the surviving evidence is directly contrary to that thesis, let's be diplomatic and say that it's "disputed" (but not very).

• Performance context. Epic was performed by a single performer to the accompaniment of a kithara, or in informal contexts, a lyre with four strings. From the late 6th century BCE onwards the most prestigious venue for performance of hexameter poetry was the competitions at the Great Panathenaia festival in Athens; after that time it became more common for a performer to recite rather than sing, holding a staff to show that he was a "rhapsode", though sometimes even then he would be accompanied by a second person on an instrument. Hexameter poetry could also be performed informally in private, or semi-publicly at symposia.

Lyric poetry is again a much more variegated mix. It was consistently sung, rather than recited (except for the episodes within Athenian drama). It could be either monodic (sung by one singer) or choral, and accompanied by a wider range of instruments. True lyric poetry, performed to a lyre, is monodic and is often rather personal in nature (most famously in the Sapphic corpus); epinician was celebratory, choral, and public, and commissioned by people for various occasions (most of the extant examples are for victorious athletes); elegy could be either, but in the course of the 6th century became the metre of choice for poems on ethical philosophy. Lyric had the same diversity of performance contexts as hexameter, but the most prestigious venues varied according to genre: the Panathenaia for epic, the Delia for choral, the Dionysia for dithyrambic, the Dionysia and Lenaia for dramatic poetry (a combo of monodic + choral), etc.

• Genre is more a matter of tone and choice of subject-matter. Homeric epic was long, grand, and serious, and dealt with heroic legend; but other hexameter poems could be short and/or light-hearted, and dealt with a wide range of subjects: wisdom poetry, cosmogony, invented oracles, hymns to gods, and antiquarian records of genealogy and similar matters.

Once again, lyric is more variegated still. In elegy, too, you can get didactic wisdom poetry and even some heroic narrative; but other lyric forms have love poetry, drinking songs, invective (either satirical or just outright insults), introspective pieces, epitaphs, songs on mythological themes that tend to go more for symbolism than epic does, mythological narratives (esp. in Stesichoros' poetry), accounts of historical events (e.g. Ion on the founding of Chios, or Timotheos on the Persian Wars): pretty much anything you can imagine.

So in general hexameter poetry had some relatively specific preoccupations, where lyric was more variegated. I wouldn't suggest thinking of the specificity of hexameter poetry in terms of constraints, though: it's more that its artificial language associated it with a certain officialness of tone.

Edit: a few minor bits of phrasing.

I just did a ton of research into this recently for a class, so here we go.

Staying away from any of the Homeric questions in this reply, Homer was alive and writing (composing might be a better word) in the 8th/9th century BCE while lyric poetry began around the 4th/5th centuries BCE. Homer's poetry actually began as an oral tradition-stories passed down through the generations- and so wasn't written down until much later, although some scenes and quotations did appear in various works of art, usually on pottery. That it was spoken is also very important to its style. Homer wrote epic poems in Dactylic hexameter, so they had a set rhythm and a sort of rhyme scheme that made them easier to remember and recite. If the bard was following the meter correctly, he would probably notice right away if he had accidentally skipped a part and could improvise to make up for the missing beats.

Lyric poetry, on the other hand, I would go so far to say is closer to what one usually thinks poetry to be. Like Homer's epic poetry, lyric poetry was made to be performed, but generally to a more reformed crowd, like at a major festival or a symposia (a sort of drinking party for the rich/intellectuals/philosophers/etc.). It was also usually accompanied by a lyre, making it closer to a song. There was no set meter or pattern that the poems had to follow, but since they were usually written down, it was okay for their structure to be more free and complex. So this also made it a lot closer to human speech, which is more flowing than the structured, almost chant or prayer like poems of Homer.

Topics are also very different between the two styles. Compare Homer's Iliad and Odyssey with Sappho's poems. Homer told whole stories (there's a reason it's called epic poetry). The Odyssey follows Odysseus from the aftermath of Troy along the long years until his return to Ithaca. While lyric poetry could tell stories, more often it was focused on more direct, human topics, things everyday people could relate to instead of some larger than life hero. Topics included death, drinking, friendship, politics, war, and love of all kinds. Sappho, a famous female lyric poet, wrote a lot of her poems on her various lovers, both male and female. But mostly female. Unfortunately, most of her works have been lost, but you can read some fragments here:

http://www.sacred-texts.com/cla/sappho/

These are just some of the main differences between the two. A lot of their differences are pretty easy to see if you compare an excerpt from Homer's work side by side to one of the lyric poets'. If you have any questions or need clarification, just ask. Most of this information comes from direct comparisons I've done between myself between samples of the two styles and a couple of lectures from my Classics professor.