Great question! It's parallel versus series circuits, as noted. In parallel, both the lamp and the LED modules get a full 3 Volts. In series, they have to share the 3 Volts, so each gets some. If it were 2 of the same kind of bulb, each would get 1/2 of the voltage. 3 bulbs in series, each gets 1/3, and so forth. The LED module makes it more complex. But let's do the biggest issue first.
Each bulb is a resistor. Connect to the intended voltage, and current flows, entirely defined by the Voltage and Resistance. Mr. Ohm noted that \$\dfrac{E}{I}=R\$, by which he meant, for our purposes, Volts divided by Amperes equals Ohms. ("E" and "I" are physicist-shorthand for charge and current, "R" is resistance.) @Spehro quoted the lamp current as 300 milliamperes, .3 Amps. 3 Volts divided by .3 Amps equals 10 Ohms: \$\dfrac{3V}{0.3A} = 10 \Omega\$.
\$\dfrac{V}{A}=R\$, multiply both sides by \$\dfrac{1}{V}: \dfrac{V}{AV}=\dfrac{R}{V}\$. Drop \$\dfrac{V}{V}\$ and you get \$\dfrac{1}{A}=\dfrac{R}{V}\$ or \$A=\dfrac{V}{R}\$. Two bulbs in series, \$10 \Omega + 10 \Omega = 20 \Omega\$. \$\dfrac{3V}{20\Omega} = .150A\$. Half the current. \$RA=V\$, \$10 \Omega \times .15 A = 1.5 V\$, half the voltage per bulb.
The LED module makes it worse because the LED has a forward voltage drop of more tha 1 Volt and 33 Ohms in series with it. So the current is \$A=\dfrac{3V - 1.xV}{10 + 33 \Omega}\$, something in the range of \$\dfrac{1.9V}{43 \Omega}\$ to \$\dfrac{1.1V}{43 \Omega}\$. At most, less than 0.05 Amperes, perhaps as little as 0.025 Amperes. Between 1/6 and 1/10 the current that the bulb got in the parallel circuit!
For discussion purposes, red LEDs have a forward voltage drop of 1.(something) volts, green are around 2 Volts, blue even higher. Of course they have finite resistance values, but it's easiest to think of them as simply removing that fixed voltage. The current can then be calculated as the remaining voltage across the resistor. If there isn't a resistor, there needs to be some more elaborate way to limit the current.
For more fun, put an appropriate size electric motor in series with a bulb, note how bright the bulb is, then put some sort of load on the motor- a finger pressing gently, a fan blade or paddle to move air, etc. See any changes?