What is the difference? Is it better to capture them?

Fighting is what happens when you get into a battle.
The outcomes of battles are:
a) you lose
b) you defeat the opponent, knocking them out
If the opponent is a Sylesti- rather than a humanoid opponent, then also:
c) you try to catch the opponent and they flee
d) you try to catch the opponent and succeed

Not sure if you've ever played pokemon, but it's pretty much the same concept.
If you catch a sylesti they'll be put into your stables. You only need to catch a sylesti if you want it, whether to keep or sell.

'Uncommon sylesti' refers to the trait point count of the pet.
Number of stars = 'difficulty', and reflects on how much Mana a pet will have when caught (for a non-fabled species, this starts at 1 star = 60 mana, up to 4 stars = 120 mana. Fabled pets start 20 points higher, 1 star = 80, capping at 3 stars = 120 (because that's the current mana cap). 5 star opponents are bosses and cannot be caught)

Colour of stars = 'rarity', and reflects how many trait points- TP- a pet has, where 1 tp is a carry, 2 tp can be 1 visible or two carries, etc.
White = common = 0tp (any trap)
green = uncommon = 1tp (any trap)
blue = rare = 2tp (blue, purple, or master traps)
purple = epic = 4tp (purple or master traps)
yellow/orange = legendary = 6tp (master trap)
red = mystical = 8tp (master trap)

What level of rarity you can find depends on what region of the maps you are in. I believe legendary and mystical only appear in the third region and the Festival Zones.

Carried (and hidden) traits will only be viewable if the pet has either been scanned with a genetic scanner or tested with a genetic test kit. People who have Exceptional Genetic Test Kits- which are infinite use- are generally willing to test pets if asked in the forums or chatboxes.