dbm -103 is a little weak signal. Generally higher the dbm value (negative), it's weaker signal. It's like logarithm value. dbm below -90 is considered strong. Above -110 dbm, 4G LTE gets very weak and may drop out to 3G any time.
Without knowing how your DNA did over there, it's hard to tell where GS4 stands in signal. Honestly I wouldn't expect GS4 do better than DNA in signal reception as DNA is one of the best reception phone on verizon. You probably didn't get GS4 because your were unhappy with DNA on signal, right?
With LTE there are a couple ways to report signal and they are significantly different from each other, there is the constant power reference signal (RSRP) which will be considerably weaker than the other way of measuring the signal (RSSI) which is the combined power of the channel rather than just the power of the reference signals. -103 db milliwatts if measuring RSRP is a fairly strong signal.
To add a bit:
RSSI is the combined power of the band as seen by the phone. This includes everything in the passband. If you are on a 10 MHz wide channel at 777 MHz then this is the power of everything from 777 to 787 MHz, including neighboring cell towers, noise, everything including the tower you are actually receiving from. Thus the strength of RSSI will be much higher than that of the reference signal and also will be higher if you are receiving on a wider channel (10 MHz vs 5 MHz).
RSRP is the average power of a reference symbol received on a single antenna port. In LTE, every third subcarrier periodically transmits a reference signal at a known power. The LTE receiving device keeps track of this, and that is a good estimation of how much power the LTE device is receiving on a single subcarrier.
RSRQ is the quality of that reference signal, how clearly it is heard over neighboring cell towers and other noise.
So the best environment would be a high RSRP and a high RSRQ (e.g., -50 dbm RSRP and -3 db RSRQ) - lots of power and it's clean, but -103 dBm RSRP if it's a high quality signal would still lead to quite good performance.
RSSI isn't a very good measure because it's just received power - it could be mostly noise.
adding a bit more:
The actual, important number I believe is hidden. CQI - channel quality indicator. This is a number from 0 to 15 which the device itself uses (or reports to the tower) that the tower can use to set it's modulation and coding. Once this value is over 10 the most complex modulation available (64QAM) is used and as the CQI increases the code rate improves (how much of the data is dedicated to error correction decreases). I'm actually not really sure why the phone should show to the user anything other than this value anyway.