I thought I'd put together something to "identify" each team. I don't know that much about playing styles being trademarks of teams or franchise players, but usually you can find something that creates an identity to a club.
Ässät
A team with a lot of tradition but less success. Even if they aren't winning, there's usually a strong sense of unity in form of the logo, the heart of spades. When there's success, it usually creates a huge boom, like last spring when they suprisingly won the championship for the first time in over 3 decades. A heated rivalry with Lukko, the neighbors.
Blues
A team that seems to be always playing second fiddle to the other two Helsinkei metropolitan area teams. Recently there's been glimpses of greatness in form of some finals appearances in the 2000's, but Blues still seems to be looking for its place in the shadow of HIFK and Jokerit.
HIFK
One of the most followed teams and the other team from capital Helsinki. Known for the always intriguing rivalry with Jokerit. Maybe the less succesful of the two, but still an established franchise with a long history. Won the whole show in 2011, before that, championships had been scarce.
HPK
One of the more lower-profile teams from a not-so-enthusiastic market. There maybe hasn't been a strong sense of identity with them during the years, but still they've silently been a good playoff performer in the last decade with some 2nd and 3rd place finishes.
Ilves
The team with arguably the most tradition in the league. Comparable to maybe the Maple Leafs in the NHL - a lot of past success and history, but little to no trophies in the last 30 years. Still the most championships of all teams, if you count the wins from the unofficial SM-sarja before 1975. A local rivalry with the other Tampere team, Tappara, is strong (although recent struggles of both teams on and off the ice have hampered the excitement).
Jokerit
The "posterboy" team of the league. The wealthiest team with the most modern arena and a team from the capital city, backed by the famous businessman, Hjallis Harkimo, ensures their place in the headlines. Even with constant high expectations, they rarely have had the top team, with most of their success coming from a 90's string of championships. Bound to move to the KHL for next season, which has created more stir.
JYP
A relatively young franchise (at least success-wise), but they've steadily become the most constant performer of the 2000's. A couple of championships and finals appearances in the past 5-6 years have lifted them to the peak of their powers. One of the perennial contenders of the today's league.
KalPa
A long history was shadowed by a 10-year spell in the second-tier league, Mestis. They made a comeback, though, and for the last couple of years, have changed from a bottom-placing club to a succesful contender, yet unable to rise to championship status. Some strong traditions with this team as well, but I don't see such a frantic following.
Kärpät
Another team who rose up the ranks to the top league after a long stint in Mestis in the turn of the millennium. The only difference is they were able to turn their potential into a dynasty of the early 2000's, winning 4 championships in 5 seasons. The northernmost club with a solid fanbase and strong finances. Known for the use of many Czech-born players.
Lukko
A similar club to their neighboring rival, Ässät: a small-town team with tradition, but not much success and a crappy arena. Recently they've made some splashes with playoff runs and a hockey boom in the smallest town in the league, but no more than that. Always creates excitement with their rivalry, though.
Pelicans
The youngest franchise, founded in 1996. Not much going for this club at all in the recent years. Usually regarded as the perennial doormat. They made a run to the finals a couple of seasons ago, but Pelicans have had a rough time gaining momentum and an identity.
SaiPa
In a similar situation as Pelicans. No success at all in the team's history, and not a strong identity, if there is one. There's always hope for improvement, though.
Tappara
The other traditional team from Tampere, with the second most succesful track record in SM-Liiga history. A long history and a rivalry with Ilves, often going head to head. The last couple of decades they've been the top dog in Tampere, with success coming more often.
TPS
The another 90's dynasty and overall the winningest team in SM-liiga history with 10 championships since 1976. Has a steady foundation in Turku, although it hasn't been all fun and games for them in the 2000's with uneven performing with highs and lows both on and off the ice. TPS has always been the club that's constantly producing home-grown stars that have been able to make the jump to North America.
Hope this helped getting you on the map. I can't say I'm 100% right with everything, but at least you probably got some idea of what's going on.